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Faculty of Sciences Department of Physics and Astronomy Search for Scalar Top Quark Partners and Parton Shower Tuning in Modelling Top Quark Pairs By Nicolas Bulté Promotor: Dr. Efe Yazgan A thesis presented for the degree of Master of Science in Physics and Astronomy June 2016 Acknowledgements To start with, I would like to express my gratitude towards Prof. Dirk Ryckbosch for his efforts into bringing me towards the point where I finalise my studies. Five years ago, when I was unsure whether physics was a good fit for me, he told me that all it takes is a decent dose of enthusiasm and motivation. Bearing this in mind I took on the challenge and ended up in my final year, writing a thesis within the field of elementary particle physics. I was given the unique opportunity to stay and work at CERN gaining valuable skills and experience, from which I will be able to profit from for the rest of my life. Further, I want to thank Prof. Didar Dobur for her endless patience and efforts into teaching me how experimental particle physics is conducted in practice. Also for the many hours she spent explaining me difficult concepts, providing me with feedback and learning me how to perform better as an experimental physicist. Next, I want to thank Dr. Efe Yazgan for always being there when I was in need of assistance, for his thorough help, feedback and guidance and for making my research count. He taught me how scientific research is conducted within an international setting which made writing this thesis an invaluable experience. Another big thanks go to Michael Sigamani, Ward Van Driessche and Antoine Pingault for taking me in their office and allowing me to see the behind the scenes work of scientific research. I’ve had a great time there and I truly enjoyed our talks and laugh bursts. Finally, I want to thank my family for their support during the past five years. For keeping up with me through the stressful periods and for providing me with the opportunities to take on this course of study. 1 Contents 1 The Standard Model 1.1 Fundamental Particles . . . . . . . . 1.1.1 The Fermions . . . . . . . . . 1.1.2 The Bosons . . . . . . . . . . 1.2 Fundamental Interactions . . . . . . 1.2.1 The Electromagnetic Force . . 1.2.2 The Weak Force . . . . . . . . 1.2.3 The Strong Force . . . . . . . 1.2.4 Electroweak Unification . . . 1.2.5 Parity and Parity-Violation . 1.3 The Brout-Englert-Higgs-Mechanism 1.4 Limitations of the Standard Model . 2 Supersymmetry 2.1 General introduction . . . . . . . . . 2.2 Supermultiplets . . . . . . . . . . . . 2.3 The Minimal Supersymmetric Model 2.3.1 Particle Categorisation . . . . 2.3.2 Broken Symmetry . . . . . . . 2.3.3 R-parity . . . . . . . . . . . . 2.4 SUSY-searches at the LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 5 6 7 7 8 10 13 15 16 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 24 25 26 26 28 29 30 3 The CMS Experiment at the LHC 3.1 The Large Hadron Collider . . . . . . . . 3.2 The CMS-Experiment . . . . . . . . . . 3.2.1 The Tracker . . . . . . . . . . . . 3.2.2 The Electromagnetic Calorimeter 3.2.3 The Hadronic Calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 31 35 36 37 38 2 . . . . . . . 3.2.4 3.2.5 3.2.6 The Superconducting Solenoidal Magnet . . . . . . . . . . . . . The Muon System . . . . . . . . . . . . . . . . . . . . . . . . . Triggering and Data Acquisition . . . . . . . . . . . . . . . . . . 4 Modelling and Analysis Software 4.1 CMSSW Application Framework . . . . . . 4.2 Particle-flow Event Reconstruction . . . . . 4.3 Monte Carlo Event Generators . . . . . . . . 4.3.1 Powheg . . . . . . . . . . . . . . . . 4.3.2 Pythia . . . . . . . . . . . . . . . . . 4.3.3 Tuning of event-generators: Professor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Analysis Part I: Search for Scalar Top Quark Partners 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 MT 2 (ll) . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Analysis strategy . . . . . . . . . . . . . . . . . . . . . . 5.4 MT 2 (ll)-tail events study for the tt̄ + jets background . . 5.5 Reduction of the tt̄ + jets background . . . . . . . . . . 5.6 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 40 41 . . . . . . 43 43 44 44 45 46 47 . . . . . . 49 49 51 52 57 59 62 6 Analysis Part II: Parton Shower Tuning in Modelling Top Quark Pairs 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Tuning of αSISR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Estimation of uncertainty on αSISR . . . . . . . . . . . . . . . . . . . . . 6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 The renormalisation scale µR . . . . . . . . . . . . . . . . . . . . . . . 6.6 Comparison between Professor Interpolation and MC Output . . . . . . 6.7 Validation with other Top Quark Distributions . . . . . . . . . . . . . . 6.8 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 63 63 65 68 68 73 74 75 78 7 Nederlandstalige Samenvatting 7.1 Introductie . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Deel I: Zoeken naar scalaire top quark partners . . . . . 7.3 Deel II: Scherpstellen van αSISR in POWHEG+PYTHIA8 delleren van top quark paren . . . . . . . . . . . . . . . . 79 79 81 . . . . . . voor . . . . . . . . . . . . . het mo. . . . . A Professor: A tuning tool for Monte Carlo event generators A.1 Determining the response function MCb (p) . . . . . . . . . . . . . . . . A.2 Goodness of Fit (GoF) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 82 84 85 86 A.3 Error Estimation: Construction of Eigentunes . . . . . . . . . . . . . . 4 86 Chapter 1 The Standard Model This chapter provides a general outline of the general principles behind the Standard Model (SM), one of the currently most successful theories within the field of physics. The SM is a theory concerning the 3 fundamental interactions of the electromagnetic, weak and strong forces and classifies all subatomic particles known today. The SM explains a wide range of experimental results, stemming from a combined effort of scientists worldwide during the second half of the 20th century. Despite its power to explain and predict, it is incomplete. The shortcomings of the SM will be discussed at the end of this chapter. 1.1 Fundamental Particles The fundamental particles are divided in 2 groups: the fermions and the bosons, shortly introduced in the following sections. 1.1.1 The Fermions The first class of particles discussed, the fermions, have spin = 1/2. They are grouped in 3 families according to their masses. Each fermion has a corresponding antifermion. The fermions, in turn, are grouped into 2 classes, the leptons and the quarks, summarised together with some of their properties in Table 1.1. The 6 leptons are the electron (e), muon (µ) and tau-lepton (τ ) with corresponding electron- (νe ), mu- (νµ ) and tau-neutrinos (ντ ). The 6 quarks are the up (u), down (d), charm (c), strange (s), top (t) and bottom-quark (b). 5 Quarks carry apart from electric charge, also a colour charge. Experimentally it is observed that no free colour states exist in nature. This means that quarks (nor gluons, to be explained later) can propagate freely and cannot be observed singularly. Quarks can be bound together forming so-called hadrons (protons, neutrons, pions, ...). The process of forming hadrons is called hadronisation. Leptons on the other hand do not carry colour charge and can certainly be found to propagate freely. 1.1.2 Table 1.1: Tabular overview of the fermions comprised within the Standard Model. The subscripts L and R stand for the left- and right handed components, T is the weak isospin and T3 its third component [1]. The Bosons Apart from gravitation (which is not comprised within the Standard Model), 3 other fundamental interactions are known. Each of those is mediated by the exchange of one or more vector bosons, which are spin = 1 particles. They are together with their exchange bosons listed below: • The strong interaction: 8 gluons (g) coupling to colour charge. • The electromagnetic interaction: the photon (γ) coupling to electric charge. • The weak interaction: the W ± -bosons and the Z 0 -boson coupling to weak charge. As can be seen from Table 1.1, the quarks carry colour-charge, allowing them to couple to themselves. The W bosons and Z bosons can also couple to themselves, as they carry weak charge. The photon does not share this property as it doesn’t carry electric charge. The only scalar particle, the Higgs-boson, requires some special attention and will be discussed in section 1.3 concerning the Brout-Englert-Higgs Mechanism. 6 1.2 Fundamental Interactions The SM is described by a Lagrangian, which describes the whole dynamics of the system. The symmetry group representing the SM Lagrangian is SU(3)×SU(2)×U(1). A symmetry group defines under which transformations a theory is invariant. According to Noether’s theorem a conservation law is associated to a physical system for each continuous symmetry of its action [2]. For the SM, the conserved quantities are colour SU(3), weak isospin SU(2) and electric charge U(1). As was discussed in Sec. 1.1.2, the (gauge-)bosons mediate the SM interactions. In general, gauge-bosons appear as virtual particles between two interaction vertices (as can be schematically depicted by Feynman-diagrams). Depending on which interaction (and the energy-scale at which the interaction occurs) a coupling strength is associated, denoted by α. In this section, each of those fundamental interactions will be discussed in more detail. 1.2.1 The Electromagnetic Force The electromagnetic force is the one that acts between electrically charged particles and is fully described by the relativistic quantum field theory of quantum electrodynamics (QED), with abelian symmetry group U(1). The gauge-boson mediating the electomagnetic interaction is the massless and electrically neutral photon γ, allowFigure 1.1: A QED process. ing the interactions up to an infinite range (the forces between particles grow weaker with the distance between them). The photon can interact with every particle carrying an electric charge Q (as can be seen from Fig. 1.1 in which two electrons with Q = -1e interact, exchanging a photon). The coupling strength of the electromagnetic interaction has the 1 [3]. Thanks to the coupling being smaller than 1, this invalue of αem (Q2 = 0) ≈ 137 teraction allows calculations in perturbation theory. It needs to be noted still that αem is a running coupling constant, meaning it depends on the value of transferred momentum Q2 . This is a consequence of the fact that the photon isn’t capable of undergoing self-interaction, as it doesn’t carry any electric charge. This can be understood with the picture that every charged particle is surrounded by a cloud of virtual γ’s and e+ e− pairs which result in a charge screening effect. The higher the transferred momentum, the closer one probes the target under study and the fewer charge screening remains. This results in a higher value of αem . 7 1.2.2 The Weak Force The weak interaction is mediated by 3 gauge bosons: the electrically neutral Z 0 boson with mass MZ 0 = 91.2 GeV/c2 , and the electrically charged W + and W − bosons with MW ± = 80.4 GeV/c2 . Due to the large masses of the exchange bosons, the weak interaction has a very short range. This allowed Enrico Fermi in the 1940s to develop a theory of interaction described as a point interaction for weak Figure 1.2: A weak interaction pro- processes. This was implemented in a quancess. tum field theory based on a SU(2) symmetry. The quantum number that governs the weakly interacting particles is the weak isospin. It is the third component, T3 , that becomes relevant because it is conserved between initial and final states. The left-handed fermions have non-zero T3 : the up-type fermions, have T3 = + 21 and all down-type fermions have T3 = − 21 .1 All right-handed fermions have T3 = 0. If one assigns T3 (W ± ) = ±1, then from Fig. 1.2 one can see that weak isospin is indeed conserved. The coupling strength of the weak interaction is 4 to 5 orders of magnitude smaller than αem at low momentum transfers. Another important artifact of the weak interaction is the following. Neutral currents through Z 0 -exchange couple between quarks of the same flavour whereas charged currents through W ± -exchange allow coupling between different generations. For leptons, both neutral and charged currents couple to particles inside the same generation. The behaviour for quarks can be understood in terms of the mixing mechanism. When a quark is produced as a mass eigenstate in a certain generation, it decays via the weak interaction as a weak eigenstate which is a superposition of quarks of all three generations. Mathematically the mixing mechanism is described by the Cabibbo-KobayashiMaskawa (CKM) matrix, which contains the weak eigenstates in terms of the mass eigenstates, as can be seen from Eq. 1.1. 0 d d Vud Vus Vub d s0 = VCKM · s = Vcd Vcs Vcb · s (1.1) 0 t t Vtd Vts Vtb t 1 Anti-fermions get the same T3 -value as their fermion counterparts. 8 An interesting type of fermions are the neutrinos (see Sec. 1.1.1) as they only interact through the weak force and gravity2 , but are not effected by electromagnetic nor strong forces. Their existence was postulated by Wolfgang Pauli in 1930 to explain the occurrence of β-decay n0 → p+ + e− + ν̄e . (1.2) The combined energies of the proton p+ and electron e− do not add up to the energy of the neutron n0 . Therefore Pauli proposed the existence of an electrically neutral neutrino νe in order for energy conservation to be satisfied. Note that also spin conservation is now satisfied, where the anti-neutrino is right-handed and the electron left-handed3 . In 1956, β-capture was proposed to detect neutrinos. Antineutrinos created in a nuclear reactor from β-decay interact with protons according to ν̄e + p+ → n0 + e+ . The positron directly annihilates with an electron forming two photons and the neutron can interact with a nucleus, also releasing a photon. This signature is unique for an antineutrino interaction [5]. Later, with the discoveries of the muon and tau-lepton, it was known that neutrinos come in three flavours (νe , νµ , ντ ). Neutrinos can be produced in various ways apart from nuclear reactors. Most neutrinos on Earth originate from nuclear processes inside the Sun (pp chain, 7 Be, 8 B, etc. [6]). Another very intensive source of neutrinos is a supernova which was measured from the SN 1987A supernova in 1987. It is also thought that a large amount of neutrinos must be present in the Cosmic Neutrino Background (CνB), a remnant from the Big Bang. Due to their incredibly low energies however, the CνB has not yet been observed. Neutrino-experiments (e.g. IceCube, Soli∂, ...) are widely conducted around the globe: they provide gateways to understand more about the cosmos (from the structure of the universe to dark matter) and provide opportunities to acquire a more thorough understanding about physics beyond the standard model (neutrino-oscillations, reactor anomaly, etc.). 2 Neutrinos were recently proven to have a small mass, see the paragraph in Sec. 1.4 on neutrinooscillations, a phenomenon not predicted by the SM. 3 A right-handed particle has its spin in the direction of its momentum whereas a left-handed particle has its spin in the opposite direction to its momentum. One often also speaks of respectively helicity h = +1 and h = −1. The helicity of the neutrino was determined by Goldhaber in 1958 [4]. 9 1.2.3 The Strong Force The strong force acts between particles carrying colour charge and is described in the framework of quantum chromodynamics (QCD) with nonabelian symmetry group SU(3). The mediating bosons are called the gluons that have spin 1 and couple to colour charge, a conserved quantity for the strong interaction. The 3 possible types of colours are green (g), red (r) and blue (b) and Figure 1.3: A QCD process. their respective anticolours. Each type of quark and gluon carry a non-zero colour charge. A quark carries one of 3 colours whereas a gluon carries a colour-anticolour combination. In total there are 8 gluons. Gluons carry colour charge themselves, allowing them to emit other gluons. This property, which is absent in QED, yields the following picture. When the momentum transfer Q2 increases, gluons start to fluctuate more and more in secondary gluons and cause an anti-screening effect [1]. Because of this, at close distances, the gluon sees less colour charge. This means that the coupling strength grows weaker for higher energy-scales that explains the running of αs . In the limit Q2 → ∞, quarks can be considered to be free due to the very low interquark coupling. This limit is called asymptotic freedom. At large distances however, the interquark coupling increases so strongy that it’s impossible to detach quarks from hadrons, this principle is called confinement and can be understood from a fluxtube being formed between two particles holding colour charge (Fig. 1.4). Figure 1.4: Left: Electromagnetic field lines for two separated electrically charged particles. The further apart the particles go, the weaker the field gets (V (r) ∼ 1/r). Right: Strong interaction field lines in the form of a flux tube. The further apart the particles go the stronger the field gets (V (r) ∼ r) [1]. 10 Confinement originates from the potential to rise as V (r) ∼ r which means that the longer the fluxtube gets, the more energy is stored in the field. Eventually the fluxtube breaks and creates new q q̄ pairs. From renormalisation group theory in QCD [7], it follows that the strong coupling constant αS at a particular scale µ1 can be written into an equation in which it is linked to itself at another scale µ2 αS (µ21 ) = αS (µ22 ) 2 µ1 µ22 1 − αS (µ22 )β1 ln (1.3) With β1 given by 11nc − 2nf (1.4) 12π in which nc = 3 denotes the possible number of colour charges and nf the number of active quark flavours (this depends on the chosen scale). Here, only one loop is taken into account as can be seen from the computation of β1 in the renormalisation group equation (with aS = απS ) β1 = das = β1 a2s + β2 a3s + ... (1.5) dµ It is this equation that for one loop corrections can be analytically transformed into Eq. 1.3. If one now chooses the scale µ2 = ΛQCD such that Eq. 1.3 becomes infinite (or the denominator becomes 0) we find the following relationship β(as ) = −µ ΛQCD = µ1 e − 1 2β1 αS (µ2 1) (1.6) That way, Eq. 1.3 can be rewritten as αS (µ2 ) = β1 ln 1 µ2 (1.7) Λ2QCD The constant ΛQCD is the scale at which perturbation theory breaks down. Asymptotic freedom is also evident from the above equation, for µ → ∞ the coupling constant αS vanishes. An overview of different measurements of αS at the scale of the mass of the Z boson MZ is given in Fig. 1.5. The current world average is αS (MZ2 ) = 0.1184 ± 0.0007 [8]. 11 36 9. Quantum chromodynamics τ-decays Baikov Davier Pich Boito SM review structure e+e- annihilation functions ABM BBG JR NNPDF MMHT lattice HPQCD (Wilson loops) HPQCD (c-c correlators) Maltmann (Wilson loops) JLQCD (Adler functions) PACS-CS (vac. pol. fctns.) ETM (ghost-gluon vertex) BBGPSV (static energy) ALEPH (jets&shapes) OPAL(j&s) JADE(j&s) Dissertori (3j) JADE (3j) DW (T) Abbate (T) Gehrm. (T) Hoang (C) electroweak precision fits hadron collider GFitter CMS (tt cross section) Figure 9.2: Summary determinations of αs (MZ2 ) offrom the sub-fields Figure 1.5: Summary of the of results various measurements αS (M world Z ). sixThe 2 discussed text. The yellow shaded) and dashed linesand indicate average αS (MZin) the = 0.1184 ± 0.0007 is (light indicated withbands the black dotted line grey the pre-average uncertainty bandsvalues [9]. of each sub-field. The dotted line and grey (dark shaded) band represent the final world average value of αs (MZ2 ). 12 whereby the dominating contributions to the overall error are experimental (+0.0017 −0.0018 ), from +0.0013 parton density functions (−0.0011 ) and the value of the top quark pole mass (±0.0013). February 10, 2016 16:30 With the concepts of asymptotic freedom and confinement introduced, for calculations one cannot ignore the very important principles of factorisation. The general problem addressed by factorisation is how to calculate high-energy cross sections [10]. A general cross section contains both short range ( Q2 ) and long range ( Q2 ) contributions. The long range part cannot be calculated perturbatively in QCD. Factorisation allows us to derive a prediction for those cross sections by separating long from short range behaviour in a systematic way. A general cross section, e.g. σpp→tt̄ , can be written as Z 1 X Z 1 dx2 fiPDF (x1 , µ2F )fjPDF (x2 , µ2F )σij→tt̄ (µ2R ) (1.8) dx1 σpp→tt̄ = i,j=partons 0 0 It consists of summing over all possible partons inside the protons that can give rise to the hard scattering process, and integrating over the momentum fractions x1,2 . The integrandum consists of two Parton Density Functions (PDFs) for each of the two interacting partons i and j. They represent the probability of finding parton i or j carrying a momentum fraction xi,j of the proton, when the proton is probed at a scale µF . The PDFs are non-perturbative, process-independent functions that can be obtained from various measurements. To find the PDF at other scales, one can be derived from the other using the DGLAP evolution functions [11]. The integrandum also contains the cross section that only takes the hard scattering into account between both partons and can be written in a perturbative expansion in the running coupling constant αS [12]. Measuring the tt̄ cross-section provides good tests of perturbative QCD and our understanding of the underlying (perturbative) hard scatter cross section. For this however, we need to have reliable PDF sets f PDF (x, Q2 ). Those are acquired from measurements in deep inelastic electron, muon, neutrino and antineutrino scattering experiments and collider data. Dedicated groups have constructed various PDF sets, e.g. NNPDF [13], CTEQ5 [14], CT10, [14], etc. 1.2.4 Electroweak Unification In the 19th century, Maxwell developed a formalism in which both electricity and magnetism can be described by one framework called electromagnetism. A similar unification was done in the 1960s by Weinberg, Salam and Glashow who combined electromagnetism and the weak interaction into one picture. This electroweak unification will be explained in this section. 13 Earlier was mentioned how the third component of weak isospin T3 of the W + and W − bosons are +1 and -1 respectively. A third state should therefore exist with T = 1, T3 = 0. This state is denoted by W 0 and forms together with W + and W − the weak isospin triplet. One further postulates the existence of a singlet state B 0 with T = 0, T3 = 0. The W 0 nor the B 0 are found in nature, they are mathematical constructs that form two other known neutral gauge bosons: the photon and the Z 0 . The idea of electroweak unification is to describe the photon and Z 0 as two orthogonal linear combinations of the W 0 and B 0 . This is expressed through the so-called electroweak mixing angle θW : |γi = cos(θW ) B 0 + sin(θW ) W 0 0 Z = −sin(θW ) B 0 + cos(θW ) W 0 If we take a look at the dominant Z 0 boson decay modes [8]: Z 0 → e+ e− µ+ µ− τ +τ − ν̄e,µ,τ νe,µ,τ (3.363 ± 0.004)% (3.366 ± 0.007)% (3.370 ± 0.008)% (20.00 ± 0.06)% we can see how the decay probability into charged leptons is significantly different from the decay probability to neutrinos. Hence the Z 0 cannot just be a neutral W 0 state with equal couplings to all fermions. Its coupling clearly depends on the electric charge, and this is a direct consequence of the fact that the Z 0 has a small electromagnetic component. 14 1.2.5 Parity and Parity-Violation An important quantity in subatomic physics is parity P. A parity transformation is defined by x −x P : y 7→ −y (1.9) z −z in which the spatial coordinates x, y and z flip their signs. It is a quantity that is conserved by electromagnetism, the strong interaction and gravity, but it is violated by the weak interaction. Parity (or P-) violation was proposed by T.-D. Lee and C.-N. Yang and experimentally proven by the Wu experiment conducted by C.S. Wu. ~ field nuclei were used. These undergo In the Wu experiment, polarised 60 Co in a B a three stage decay process; namely a nuclear β-decay followed by two high energy photons according to 60 Co(5+ ) → 60 Ni∗∗ (4+ ) → 60 Ni∗∗ (2+ ) → Ni∗∗ (4+ ) + e− + ν̄e 60 Ni∗∗ (2+ ) + γ 60 Ni∗∗ (0+ ) + γ 60 Weak Electromagnetic Electromagnetic the numbers between parentheses denote the spin-parity J P . In the experiment, photons are counted at angles 0◦ and 90◦ with respect to the principal axis along the polarisation direction. The photon emission is anisotropic and this anisotropy serves as a measurement of the polarisation: γ = (W (90◦ ) − W (0◦ ))/(W (90◦ ) + W (0◦ )). If γ = 0, then the nuclei have lost all polarisation. The electron counting rate We (θ) is measured and should be equal if parity is conserved by weak interactions for We (0◦ ) ~ field. and We (180◦ ). Measuring this can be achieved by flipping the B The observation by Wu et al. [15] was that electrons are more likely to be emitted opposite to the field than along it, proving that parity is violated by the weak interaction. Moreover, it seems that parity is maximally violated, leading to the V − A interaction theory proposed by Feynman and Gell-Mann [16]. 15 1.3 The Brout-Englert-Higgs-Mechanism The Brout-Englert-Higgs-Mechanism was named after the three physicists Robert Brout, François Englert and Peter Higgs, based on their work in the 1960s [17]. This mechanism is essential to explain the generation mechanism of mass for the W and Z gauge bosons, while the photon remaining massless. Furthermore, it predicts the existence of a massive scalar, spin-0 particle: the Higgs boson. As the Higgs boson was eventually discovered at the LHC by the CMS and ATLAS experiments in 2012, it deserves a deeper study of it’s theoretical backgrounds. Out of symmetry considerations, the weak force’s gauge bosons should have zero mass, which clearly is not the case. This conundrum is fixed through introduction of a quantum field, the Higgs field, which stretches throughout all space. The masses of the W and Z bosons will arise from breaking the SU(2)L × U(1)Y local gauge symmetry of the electroweak sector of the Standard Model. This will be more elaborately discussed below, where most of the discussion is based on [18]. Consider a scalar field φ with the potential 1 1 V (φ) = µ2 φ2 + λφ4 2 4 with the corresponding Lagrangian (1.10) 1 1 1 L = (∂µ φ)(∂ µ φ) − µ2 φ2 − λφ4 . (1.11) 2 2 4 1 µφ2 represents the mass of the particle, the 41 λφ4 represents a quartic coupling of the 2 field φ with coupling strength 14 λ. The vacuum state is the lowest energy state of the field φ and corresponds to the minimum of the potential. It has its minima at r −µ2 φ = ±v = ± (1.12) . λ If one considers µ2 < 0, the field has a non-zero vacuum expectation value. There are two possible vacuum states: φ = +v and φ = −v. The choice of one of the vacuum states breaks the symmetry of the Lagrangian, this is called spontaneous symmetry breaking. If one makes an arbitrary choice of the vacuum state to be φ = +v, small perturbations of the field around the vacuum state can be considered by writing φ(x) = v + η(x). The Lagrangian now becomes 16 1 1 1 L(η) = (∂µ η)(∂ µ η) − µ2 (v + η)2 − λ(v + η)4 . (1.13) 2 2 4 With the minimum of the potential given by µ2 = −λv 2 , the expression can be written as 1 1 1 L(η) = (∂µ η)(∂ µ η) − λv 2 η 2 − λvη 2 − λη 4 + λv 4 . 2 4 4 Comparing this with the Klein-Gordon equation for spin-0 scalar particles 1 L = (∂µ φ)(∂ µ φ) − 2 it becomes clear that the term proportional to η 2 mη = √ 2λv 2 = 1 2 2 mφ 2 represents a mass (1.14) (1.15) p −2µ2 and therefore the Lagrangian 1.15 in terms of the excitations about the minimum describes a massive scalar field. If we apply the same principles of spontaneous symmetry breaking to a complex scalar field, more interesting properties emerge. With a field 1 φ = √ (φ1 − iφ2 ) 2 (1.16) the Lagrangian becomes 1 L = (∂µ φ)∗ (∂ µ φ) − µ2 (φ∗ φ) + λ(φ∗ φ)2 2 1 1 1 1 = (∂µ φ1 )(∂ µ φ1 ) + (∂µ φ2 )(∂ µ φ2 ) − µ2 (φ21 + φ22 ) − λ(φ21 + φ22 )2 . 2 2 2 4 (1.17) (1.18) This Lagrangian is invariant under φ → φ0 = eiα φ and therefore possesses a global U(1) symmetry. Depending on the sign of µ2 , the potential has a different shape. For µ2 < 0 it has an infinite set of minima denoted by the dashed circle line in Fig. 1.6 (b). Nature will choose one of the vacuum states, causing to spontaneously break the U(1) symmetry. If one chooses a vacuum state (φ1 ,φ2 ) = (v,0) and expands the scalar field φ as φ1 (x) = η(x) + v and φ2 (x) = ξ(x) one gets for φ and the Lagrangian L 1 φ = √ (η + v + iξ) 2 1 1 L = (∂µ η)(∂ µ η) + (∂µ ξ)(∂ µ ξ) − V (η, ξ) 2 2 17 (1.19) (1.20) Figure 1.6: The potential V (φ) = µ2 (φ∗ φ) + λ(φ∗ φ)2 for two different values of µ2 : (a) µ2 > 0 and (b) µ2 < 0. with 1 1 1 1 (1.21) V (η, ξ) = − λv 4 + λv 2 η 2 + λvη 3 + λη 4 + λξ 4 + λvηξ 2 + λη 2 ξ 2 4 4 4 2 Every quadratic term in the field can be interpreted as a mass whereas all higher order terms are interaction terms. This allows the Lagrangian to be rewritten as 1 1 1 L = (∂µ η)(∂ µ η) − m2η η 2 + (∂µ ξ)(∂ µ ξ) − Vint (η, ξ). (1.22) 2 2 2 √ This Lagrangian represents a scalar field η with mass mη = 2λv 2 and a massless scalar field ξ. This can be understood by looking at the potential in Fig. 1.6 (b), in which it’s clear that the excitations η are up the potential walls and those of ξ are along the dashed circle where the potential doesn’t change. The massless scalar particle ξ is known as the Goldstone boson. For the complex scalar field, the Lagrangian is not invariant under a local U(1) gauge transformation φ(x) → φ0 (x) = eigχ(x) φ(x). (1.23) This can however be solved by introducing the replacement of the derivatives in the Lagrangian by the corresponding covariant derivatives 18 ∂µ → Dµ = ∂µ + igBµ . (1.24) L = (Dµ φ)∗ (Dµ φ) − V (φ2 ) (1.25) Then the Lagrangian, is gauge invariant if the new gauge field Bµ transforms as Bµ → Bµ0 = Bµ − ∂µ χ(x) (1.26) F µν = ∂ µ B ν − ∂ ν B µ (1.27) If we take into account that We find after plugging Eq. 1.20 into the Lagrangian 1.25, that 1 1 1 1 L = (∂µ η)(∂ µ η) + (∂µ ξ)(∂ µ ξ) − Fµν F µν + g 2 v 2 Bµ B µ − Vint + gvBµ (∂ µ ξ) (1.28) 2 2 4 2 There is a problem with the last term in the above expression, which is a direct coupling between the spin-1 gauge field B and the Goldstone field ξ. This can be eliminated by performing the gauge transformation Bµ (x) → Bµ0 (x) + 1 ∂µ ξ(x). gv (1.29) The Lagrangian becomes 1 1 1 L = (∂µ η)(∂ µ η) − λv 2 η 2 − Fµν F µν + g 2 v 2 Bµ0 B 0µ −Vint |2 {z } |4 {z2 } massive η (1.30) massive gauge field We notice that the Goldstone field ξ has disappeared. The original gauge transformation of φ(x) becomes φ(x) → φ0 (x) = e−ig ξ(x) gv φ(x) = e−iξ(x)/v φ(x). (1.31) In first order, the form φ(x) = √1 [v + η(x)] eiξ(x)/v can be rewritten as (2) 1 φ(x) ≈ √ [v + η(x)] eiξ(x)/v 2 This yields 19 (1.32) 1 1 φ(x) → φ0 (x) = √ e−iξ(x)/v [v + η(x)] eiξ(x)/v = √ (v + η(x)). 2 2 Choosing the field φ(x) to be entirely real, i.e. (1.33) 1 1 φ(x) = √ (v + η(x)) ≡ √ (v + h(x)) (1.34) 2 2 corresponds to the choosing the previously discussed gauge; the unitary gauge. We now notice that the only physical field in the unitary gauge is the remaining Higgs-field h(x). The Goldstone field ξ(x) was “eaten” by the gauge field. If we write µ2 = −λv 2 , the Lagrangian can be finally rewritten as 1 1 1 1 L = (∂µ h)(∂ µ h) − λv 2 h2 − Fµν F µν + g 2 v 2 Bµ B µ + g 2 vBµ B µ h2 − λvh3 − λh4 . {z } | | |2 {z } |4 {z2 } interactions {z 4 } h, B massive h massive gauge boson self-interaction h (1.35) This Lagrangian describes a massive scalar Higgs field h and a massive gauge boson B, obeying a U(1) local gauge symmetry. We see thus how the Higgs-mechanism effectively generates mass for the gauge bosons and predicts the existence of a massive scalar particle, the Higgs Boson. In the Salam-Weinberg model, the mass terms are found by writing the Lagrangian such that it respects the SU (2)L × U (1)Y local gauge symmetry of the electroweak theory by replacing the derivatives by the appropriate covariant derivatives. We omit the calculation and give the results below. The mass terms for the W (1) and W (2) spin-1 fields appear as: 1 1 1 mW Wµ(1) W (1)µ and mW Wµ(2) W (2)µ with mW = gW v (1.36) 2 2 2 The W (3) - and B-terms couple together in the Lagrangian and mix. The relationship between the physical fields and the underlying ones in the Lagrangian is Aµ = cosθW Bµ + sinθW Wµ(3) with mA = 0 q 1 2 with mZ = v gW Zµ = −sinθW Bµ + + g 02 2 Where Aµ represents the massless photon, and Zµ the massive Z-boson. The BroutEnglert-Higgs-Mechanism thus effectively predicts the masses of the gauge bosons, as well as the existence of a massive scalar spin-0 boson. After the discovery of the Higgs boson in 2012 at the CERN LHC, François Englert and Peter Higgs were awarded the 2013 Nobel Prize in physics. cosθW Wµ(3) 20 The Brout-Englert-Higgs mechanism can be used to generate the masses for the fermions. Without giving the exact theoretical considerations, a new interaction, the Yukawa interaction, is introduced that provides the coupling between a scalar field φ and a Dirac field ψ. Dirac fields describe the fermions with spin-1/2. The Yukawa couplings of the fermions to the Higgs field are given by √ mf (1.37) 2 v with v = 246 GeV the vacuum expectation value of the Higgs field. They occur in a Lagrangian of the form gf = mf ¯ L = −mf f¯f − ffh v 1.4 (1.38) Limitations of the Standard Model Apart from the great successes of the Standard Model, some fundamental problems still go unexplained. In essence, it is still unsure how the SM behaves at very high energy scales. As accelerators grow stronger, it might become possible to hit up till now unexplored frontiers and new physics start emerging. An overview of the most important indicators that the SM is incomplete, and the challenges it poses for physicists today, is listed below. General relativity The Standard Model does not incorporate the fourth fundamental force: the gravitational interaction. In fact, the SM has to become compatible with general relativity, and this is not the case. Neutrino-oscillations In the early 2000’s the Super-Kamiokande Collaboration [19] and Sudbury Neutrino Observatory (SNO) [20] Collaborations confirmed the occurrence of neutrino-oscillations: the changing of neutrino flavour as neutrinos travel a particular distance. In case of two neutrino generations, the probability of a neutrino να changing flavour to νβ is given by ∆m2 L 2 2 Pα→β = sin (2θ) sin (1.39) 4E with θ the mixing angle between the neutrino flavour- and mass-eigenstates, E its energy, L its distance travelled and ∆m its mass difference. This mechanism thus proves 21 that neutrinos must have a (small) mass. This phenomenon is not predicted by the Standard Model. The Hierarchy problem The so-called hierarchy problem boils down to the question why the weak force is 1032 times stronger than gravity. In another formulation, one can ask why the Higgs mass is so small as was experimentally found [21], [22]. The Higgs boson has a bare mass4 that gets corrected for each particle interacting with it [23]. The heavier the particle, the larger this correction. The top quark adds such a large correction that it is at least peculiar why the bare Higgs mass is so different from the measured one. Figure 1.7: Rotational curve of an arbitrary galaxy. A: model without dark matter. B: model with dark matter, as it is measured. Figure 1.8: Assumed energy distribution of the universe. Dark Matter and Dark Energy Dark matter and dark energy are hypothetical substances believed to account for over 96% of all energy in the universe (Fig. 1.8). Dark matter has never been directly observed, as it does not interact electromagnetically, but its existence is inferred from various gravitational effects. One of the most important reasons to believe that there is such a thing as dark matter is that it is a necessary ingredient to explain the observationally measured rotation curves of galaxies, as depicted in Fig. 1.7. It is observed that at large distances from the galaxy center, stars move at larger rotational speeds 4 The bare mass is the mass of the particle when the scale of distance to it approaches zero. The measured mass is the “dressed” mass, which is an addition to the bare mass because the particle interacts with virtual particle pairs temporarily created by the force field around the particle. 22 than can be explained from the observed mass distribution. Therefore extra matter is incorporated in the model: dark matter. Another example are the various gravitational lenses that have been observed. Those are massive objects (typically galaxies) bending the trajectories of light coming from behind and distorting the images seen through telescopes. The lens is often observed to be much less heavy than the lensing effect it causes, adding to the belief there is more mass present than can be observed. Other evidence can be found in measured anisotropies in the angular distribution of the Cosmic Microwave Background (CMB) in which cold dark matter is introduced as an explanation of this anisotropy5 . The Standard Model does not contain any particle candidate that could be explained as being the primary dark matter component, it thus only explains ∼4% of observable matter in the universe. There are models that incorporate DM particles into the SM (e.g. MACROs [25]), however those are less popular. Matter-antimatter asymmetry The observable universe is made of primarily matter and contains only small proportions of antimatter. The Standard Model predicts that matter and antimatter should have been created in (almost) equal amounts when the universe was formed. There is no mechanism incorporated within the SM that can explain the observed asymmetry. All predictions made by the SM have been confirmed, often with high precision. At the time of writing there is no single theory that successfully incorporates all the needed extensions explained above (especially general relativity, which is also highly successful). The most popular extension is Supersymmetry (SUSY) and is very intensively searched for at the LHC. A general introduction to SUSY is given in Chapter 2. 5 This is often called the ΛCDM model. Recent CMB anisotropies were measured by WMAP [24] 23 Chapter 2 Supersymmetry 2.1 General introduction As is alluded in the previous chapter, the Standard Model clearly is an incomplete theory. Up till now, no framework exists to describe the physics at very high energies up to the Planck scale, with MP = (8πGNewton )−1/2 = 2.4×1018 GeV/c2 , where quantum gravitational effects become important. This was already addressed before in a brief mentioning of the so-called hierarchy problem (Sec. 1.4). The Higgs boson receives quantum corrections to its mass originating from the virtual effects of particles that couple to the Higgs field. This is described in more detail in Ref. [26], where a large part of the discussion in this chapter is based on. Consider for example the loop diagram given in Fig. 2.1. In the Lagrangian a term of the form -λf H f¯f will originate and yields a correction (after applying the Feynman rules, which is omitted here) |λf |2 2 Λ + ... (2.1) 8π 2 UV with λf the Yukawa-coupling. The mass of a fermion is proportional to its Yukawacoupling meaning that the Higgs boson couples most strongly to the most massive particles. ∆m2H = − Figure 2.1: One-loop quantum correction to the Higgs squared mass parameter m2H . 24 The negative sign originates from the spin-statistics theorem, which dictates that fermions yield negative contributions whereas bosons yield positive contributions. The factor ΛUV is called the ultraviolet cutoff and denotes the scale up to which the Standard Model is valid. Filling the top quark into the equation yields unnaturally high corrections to MH2 at the Planck mass, since the correction depends quadratically on the cut-off. This particular problem can be solved by the principles of supersymmetry. It is an extension of the SM that creates superpartners for all Standard Model particles. In our particular example, the addition of supersymmetric partners adds extra terms to Eq. 2.1 and yields cancellation between them, effectively reducing the correction to zero. Formally, Supersymmetry or SUSY, is a symmetry that relates bosons and fermions. A supersymmetry transformation turns a bosonic into a fermionic state and vice versa. This is done through the introduction of an anti-commuting spinor with Q |Bosoni = |Fermioni Q |Fermioni = |Bosoni 2.2 (2.2) Supermultiplets The supersymmetric particles are divided in supermultiplets . Each of those contain both boson and fermion states who are superpartners of each other. Because the squared-mass operator -P 2 commutes with the operators Q and Q† it follows that particles within the same supermultiplet have the same masses. The generators Q and Q† also commute with the generators of the gauge transformations. This has the consequence that particles within the same supermultiplet must have the same electric charges, weak isospin and color charges. A final very important remark is that it can be proven from quantum mechanical considerations (done in [26]) that the number of degrees of freedom for fermions and bosons must be exactly equal within the same supermultiplet. Therefore one has nF = nB . (2.3) This property can then be used to construct two types of multiplets that are used to classify the supersymmetric particles. 25 Chiral supermultiplets The first considered supermultiplet consists of a single Weyl fermion (with two spin helicity states; nF = 2) and two real scalars (nB = 1). The two real scalar degrees of freedom are then combined into a complex scalar field. The combination of a twocomponent Weyl fermion and a complex scalar field is called a chiral supermultiplet. Gauge supermultiplets A second possible supermultiplet contains a spin-1 vector boson that is massless (in order for the theory to be renormalizable). Such a boson has 2 helicity states which yields nB = 2. Its superpartner must therefore have two degrees of freedom, nF = 2, and this is again a massless spin-1/2 Weyl fermion. The combination of spin-1/2 gauginos and spin-1 gauge bosons is called a gauge supermultiplet. In a supersymmetric extension of the SM, each of the already known fundamental particles are classified into a chiral or gauge supermultiplet. They each have a superpartner that has a spin differing by 1/2 unit. Other combinations of spins are possible but those can always be reduced to chiral or gauge supermultiplets [26]. 2.3 The Minimal Supersymmetric Model The Minimal Supersymmetric Model (MSSM) is a minimal supersymmetric extension of the Standard Model. It contains the least number of particles you can add to the Standard Model to make a viable SUSY model. A first step is to categorise the already known SM particles into the SUSY picture, then supersymmetry breaking will be discussed, and a conserved quantity called R-parity will be introduced. This section is based on Ref. [26]. 2.3.1 Particle Categorisation The Standard Model fermions (quarks and leptons) are members of chiral supermultiplets. This is because left-handed fermions couple differently under the gauge group than their right-handed parts. The supersymmetric particles of fermions are spin-0 particles. As a naming convention, the letter “s” is put in front of the SM partner’s names (“s” denotes “scalar”). This way, the sfermions are subdivided into squarks and sleptons. Their symbols are the same as those of the SM ones, but they get an extra tilde (˜). For example, the left- and right-handed selectrons are denoted by ẽL and ẽR . The smuons and staus are given accordingly by: µ̃L , µ̃R , τ̃L , τ̃R . Only left-handed SM neutrinos exist, the sneutrinos are therefore denoted by: ν̃e , ν̃µ , ν̃τ . Finally, the squarks 26 are given by q̃L , q̃R with q = u, d, s, c, b, t. Since the gauge interactions within the chiral supermultiplet are equal, also the left handed squarks couple to the W boson whereas the right handed squarks do not. The Standard Model vector bosons reside in gauge supermultiplets. The gauge interactions of the SU (3)C group of QCD are mediated by the gluon g whose supersymmetric partner is the gluino g̃. The spin-1 bosons W + , W 0 , W − and B 0 of the electroweak gauge symmetry SU (2)L × U (1)Y have the winos and bino as supersymmetric partners: W̃ + , W̃ 0 , W̃ − , B̃ 0 . As described in Sec. 1.2.4, electroweak symmetry breaking mixes the W 0 and B 0 eigenstates to give the mass eigenstates Z 0 and γ. The corresponding gaugino mixtures of W̃ 0 and B̃ 0 are the zino Z˜0 and photino γ̃. The Standard Model Higgs boson (spin-0) resides in a chiral supermultiplet. It turns out however, that one Higgs chiral supermultiplet is not enough. This would result in what is called a gauge anomaly and would cause an inconsistent quantum theory1 . It turns out that the problem can be solved by introducing two Higgs multiplets: one with Y = + 21 and one with Y = − 21 2 . They are also both necessary as only a Y = + 12 Higgs chiral supermultiplet can have the necessary Yukawa couplings to give masses to charge +2/3 up-type quarks. The Y = − 21 Higgs chiral supermultiplet in its turn can only provide masses for the -1/3 down-type quarks (and to the charged leptons). The scalar fields with Y = + 12 and Y = − 12 are named Hu and Hd , respectively. The weak isospin components of Hu with T3 = (1/2, −1/2) have electric charges 1,0 and are denoted by (Hu+ ,Hu0 ). In a similar way we have (Hd0 ,Hd− ). The superpartners then become H̃u+ ,H̃u0 and H̃d0 ,H̃d− and are referred to as the Higgsinos. The higgsinos and electroweak gauginos mix with eachother because of electroweak symmetry breaking. The neutral higgsinos (H̃u0 , H̃d0 ) and the neutral gauginos (B̃, W̃ 0 ) combine to form four mass eigenstates called neutralinos (χ̃01 , χ̃02 , χ̃03 , χ̃04 ). The charged higgsinos (H̃u+ , H̃d− ) and winos (W̃ + , W̃ − ) mix to form two mass eigenstates with charge − + − ±1 called the charginos (χ̃+ 1 , χ̃1 , χ̃2 , χ̃2 ). The above discussed sparticles are all the particles comprised within the MSSM and are summarised in Tables 2.1 and 2.2. 1 Gauge anomalies invalidate the gauge symmetry of a quantum field theory. Gauge symmetries are theories where symmetry transformations are space-time dependent. They are used to generate dynamics, namely the gauge interactions. For a more thorough explanation, refer to [27]. 2 Y stands for weak hypercharge and is given by Y = 2(Q − T3 ), with Q the electric charge and T3 the third component of weak isospin. 27 Table 2.1: Chiral supermultiplets in the Minimal Supersymmetric Standard Model. The spin-0 fields are complex scalars, and the spin-1/2 fields are left-handed two-component Weyl fermions [26]. Table 2.2: Gauge supermultiplets in the Minimal Supersymmetric Standard Model. The spin-1 fields are the SM vector bosons, the spin-1/2 ones are their supersymmetric counterparts [26]. 2.3.2 Broken Symmetry In Sec. 2.2 it was explained how particles within the same supermultiplets must have the same mass, this however requires some extra attention. At the time of writing, not a single superpartner of the SM particles has been discovered. If supersymmetry would be an unbroken symmetry, then there would have to be selectrons ẽL and ẽR with masses exactly equal to the electron mass me = 0.511 MeV (this is also the case for all other SM particles). Those sparticles should be easily detectable but they haven’t been found. Supersymmetry must thus be a broken symmetry, causing sparticles to have higher masses than their SM counterparts. In other words, this means that the model explaining the symmetry breaking, should have a Lagrangian density that is invariant under supersymmetry, but a vacuum state that is not. 28 The exact theoretical considerations fall outside the scope of this study, and it is satisfying enough to understand that superpartners will have higher masses than their SM counterparts. The general idea however of supersymmetry breaking is to introduce an effective MSSM Lagrangian consisting of two parts L = LSUSY + Lsoft (2.4) with LSUSY containing the gauge and Yukawa interactions, preserving supersymmetry invariance and Lsoft violating supersymmetry. The second part is called soft because the breaking terms are constructed such, that they do not introduce quadratic divergences [28]. We also note that there are no theoretical predictions towards the actual masses of SUSY particles, which makes SUSY searches extra challenging. 2.3.3 R-parity R-parity RP [29], [30] is a discrete multiplicative symmetry that can be written as RP = (−1)3B+L+2S , (2.5) with B denoting the baryon number, L the lepton number and S the spin of a particle. It has the property that all SM fields have RP = +1 whereas all the superpartners of SM fields have RP = −1. R-parity is introduced into the MSSM in order to eliminate the possibility of B- and L-number violation. It is mentioned here as R-parity conservation3 has three very important consequences: • SUSY particles are always produced in pairs. • The lightest sparticle with RP = −1, called the lightest supersymmetric particle or LSP, must be stable4 . Considering it to be massive, weakly interacting and electrically neutral, makes the LSP a viable candidate to explain dark matter. It is clear that R-parity conservation puts some interesting constraints on the possible SUSY Feynman-diagrams. Also note that the LSP behaves like a neutrino and thus escapes detection, it is in general identified with the lightest neutralino χ̃01 . 3 R-parity conservation is an assumption made in the MSSM, theories with R-parity violation exist too, but aren’t considered here. 4 Because of energy- and R-parity conservation it cannot decay to other SUSY or SM particles. 29 2.4 SUSY-searches at the LHC Due to the fundamental differences between the SM and SUSY, it deserves our attention to understand how SUSY-searches at the LHC can be conducted. From theoretical calculations, it seems that squark and gluino production are expected to be dominant processes. Since they are expected to be heavy, they will produce long decay chains eventually leading to a final state with only SM particles and LSPs. Those Standard Model particles will typically be leptons (e or µ), neutrinos, and quarks that hadronise into jets. The LSP is neutral and interacts weakly, causing it to have the same behaviour as the neutrino in a detector in turn causing it to be undetectable. It becomes clear that one has to construct smart techniques in order for SUSY-signatures to be detectable. Obviously, typical SUSY-processes will have large ETmiss due to the presence of the LSP. This can be exploited by constructing various variables who can discriminate SUSY- from regular SM-processes (an example will be given in the Analysis section). SUSY searches are typically distinguished by the number of leptons required in the final state. We list a few possibilities below. • Zero-lepton searches typically focus on the escaping dark-matter candidate (the LSP). The cross section for events with large ETmiss , large number of jets and no leptons is very large. Unfortunately, the SM background shares this property, making this a difficult channel to look at. • One-lepton searches use the fact that SUSY models couple to weak interaction, meaning it is likely that a single lepton (+ ν) is produced. • A very SUSY-specific signal is the same-sign dilepton channel. A possible process can be understood if we consider the decays g̃ → t̃t̄ and g̃ → t̃∗ t, which are possible in simple R-parity violating models, and occur with the same probability. Gluino pair-production could lead to events with two same-sign tops. If both decay leptonically a same-sign dilepton signature is obtained. Obviously, opposite-sign dilepton searches are also conducted, this will be more elaborately discussed in Chapter 5 • Multi-lepton searches also exist in which production of charginos and neutralinos are considered, giving rise to 3 leptons. The above listing shows some of the possible SUSY-search scenarios, and more possibilities exist. This thesis will contain research work within an opposite-sign dilepton analysis in which stop-quark production is the analysed signal in the MSSM, where R-parity conservation is expected. 30 Chapter 3 The CMS Experiment at the LHC 3.1 The Large Hadron Collider The Large Hadron Collider is the biggest and most powerful particle accelerator in the world. It is located at CERN (European Organization for Nuclear Research) near Geneva, Switzerland. The accelerator consists of a circular-shaped tunnel with a circumference of approximately 27 km and allows protons (as well as heavy ions) to collide at very high center-of-mass energies. In order to establish proton collisions, two beam pipes are necessary to bend the protons in opposite directions and allow collisions at fixed points along the accelerator. The guidance and bending is done through the use of superconducting dipole magnets to bend, and quadrupole magnets to focus the protons along the optical axis. Superconductivity is necessary as extremely high currents up to 12×103 A are required in the magnet coils to provide high magnetic fields. Higher order magnets are also used to provide additional corrections to the beam. The design luminosity of the LHC is L = 1034 cm−2 s−1 , achieved from the circulation of protons through the accelerator (about 11×103 turns per second) that are bunched together with bunch spacings of 25 ns. Protons are √ currently being accelerated to 6.5 TeV each, resulting in a center-of-mass energy of s = 13 TeV (before this, the LHC also operated at 7 and 8 TeV at respective peak luminosities of 3.7 × 1033 cm−2 s−1 and 7.7 × 1033 cm−2 s−1 ). This is a never-before achieved energy and provides tremendously exciting research opportunities. On the LHC-ring (schematically depicted in Fig. 3.1), four main experiments are in31 stalled around the interaction points where the beams collide, namely, ATLAS and CMS; two general purpose detectors, and ALICE and LHCb; conducting heavy-ion and b-physics experiments, respectively. This thesis will be subjected to an analysis situated within the scope of the CMS-experiment, which will be more thoroughly explained in the next section. Figure 3.1: Schematic overview of the LHC-complex. Depicted in yellow are the four c CERN Collaboration main experiments. The injection of protons into the LHC is followed after a gradual increase in their energy using different accelerator structures depicted in Fig. 3.1. The protons originate from a bottle of hydrogen gas, whose electrons are stripped after applying an electric field. They are passed to a Linear Accelerator (LINAC) structure, accelerating the protons to an energy of 50 MeV before entering the booster. The booster accelerates to an energy of 1.4 GeV, which subsequently passes the protons to the Proton Synchrotron (PS) and eventually to the Super Proton Synchrotron (SPS), accelerating respectively to 25 and 450 GeV. Finally, the protons are transferred to the two beam pipes of the LHC, further increasing their energy up to 6.5 TeV at which they collide. A schematic view of a proton-proton collision is given in Fig. 3.2. Two partons from each of the two protons interact with each other forming a primary, hard-scatter interaction vertex (depicted with the red dot). This process produces a new state that eventually decays into a spray of particles. Those undergo a showering process after 32 Figure 3.2: Schematic view of a proton-proton collision [31]. which they hadronise into the particles that finally reach the detector. The huge advantage of a proton-proton accelerator over for example an electron-positron accelerator is that a very wide energy range can be probed. This is a direct consequence of the fact that partons inside the proton carry a fraction of the momentum of the proton, thus allowing collisions to take place at different values of transferred momentum. With proton-proton collisions, the LHC allows to look at the fundamentals of what matter consists of and helps scientists to gain a deeper and more profound understanding of the fundamental open questions in physics. This concerns the basic laws governing the interactions among elementary particles, the scales at which current theories break down or not, etc. Apart from proton-proton collisions, the LHC also serves another purpose: heavy-ion collisions. Reactions between heavy-ions have proven themselves very useful regarding the investigation of the thermodynamical properties of QCD matter. The difference with protons colliding is not that single protons, but a bunch of protons and neutrons are smashed together to form a very dense nuclear matter system. With these type of collisions, physicists try to investigate the quark-gluon plasma. 33 The quark-gluon plasma is a hypothesized phase of nuclear matter, believed to exist at either very high densities or very high temperatures (or both). If one were to compress a nucleus (without altering the temperature) to very high densities (∼10×), individual nucleons would start overlapping. If one on the other hand would increase the temperature (without altering the density), the individual nucleon-nucleon interactions would increase in such a way, that the collisions between them would make it impossible to assign a quark or gluon to any hadron in particular. The state is said to consist of asymptotically free quarks and gluons. The discovery of the quark-gluon plasma could provide great confidence in our understanding of strongly interacting matter. Apart from that, it would be a simulation of the universe at a very early stage in its history (∼30 µs after the Big Bang). At CERN, the LHC operates approximately one month yearly in nuclear collision mode. Pb-nuclei collide at 2.76 TeV per nucleon pair and produce big sprays of strongly interacting particles around the central collision point who are then detected by the detectors installed on the LHC. Figure 3.3: Two heavy-ions collide with partial overlap (left). A spray of strongly interacting particles (black dots) are produced (right) [1]. 34 3.2 The CMS-Experiment The CMS, or Compact Muon Solenoid experiment, is a general purpose particle physics detector situated on the LHC-ring. It is cylindrically shaped with a diameter of 15 m and a length of 28.7 m, weighing 14000 metric tonnes. It gained world fame after it discovered together with ATLAS the Standard Model Higgs-Boson in July 2012. The main objectives of the CMS-experiment are (1) to explore physics at very high energy scales (TeV-range), (2) to test the SM at new energy regimes and phase space regions that have not been explored before, (3) to increase the precision of SM electroweak and QCD parameters, (4) to study the properties of the recently discovered Higgs-boson, (5) to look for physics beyond the standard model, in particular Supersymmetry and (6) to help in providing a possible explanation of dark matter. The detector has an onion-like structure where each layer serves a different purpose that helps in identification of particles originating from the central collision. The signal from each detector component is then combined in order to reconstruct the original process that took place. This is not an easy task as collisions are happening at a frequency of 40 MHz, making extremely fast electronics and smart algorithms necessary to distinguish the different interactions. The different components of the detector are described below. c CMS Collaboration Figure 3.4: Overview of the different sections of CMS. 35 3.2.1 The Tracker The tracker sits at the very core of the detector and surrounds the interaction point. It has a length of 5.8 m and a diameter of 2.5 m and covers a pseudorapidity range of |η| < 2.51 . The tracker’s function is to provide a precise measurement of the trajectories of charged particles emerging from the interaction point as well as identifying secondary vertices from prolonged b quark and τ -decays. Due to the extremely intense dose of radiation at which the tracker is exposed, it is composed entirely from silicon-based detector technology, as this material provides a very long lifetime. The tracker consists of two main sections. The most central part is the pixel detector consisting of three co-axial cylindrical layers of pixel detector modules, closed of at the side by another two pixel modules. The pixel detector delivers three high-precision space points for each particle trajectory (charged particle trajectories are curved due to the presence of a high magnetic field, to be explained later). It is this precision that allows accurate measurements of secondary vertices as displayed in Fig. 3.5. In order to detect b quarks (typically necessary for detecting processes as depicted in Fig. 4.3.1) one determines the impact parameters d (the distance of closest approach of a secondary track to the primary vertex) and determines whether they’re larger than some reference to be sure the track originated from a secondary vertex. A distribution of Lxy (the decay distance in the plane transverse to the beam direction) allows comparison with typical b quark lifetime distributions. Figure 3.5: Display of a secondary vertex originating from b-quark decay. d and Lxy c CDF Collaboration are also depicted. The second and outermost section of the track system is the silicon strip tracker and 1 The pseudorapidity η is a measure for the angle of a particle relative to the beam-axis and is defined as η = −ln tan θ2 . It is frequently used as η is boost-invariant, whereas θ isn’t. This allows displays of particle energy depositions in so-called lego plots between the two boost invariant quantities η and φ, with the latter being the (boost-invariant) angle in the transverse plane. This way, angular differences are not influenced by the boost of the particles. 36 provides additional resolution to the tracks measured. After the particles leave the pixel detector, they pass through 10 layers of silicon strip detectors, reaching out to a radius of about 1.2 m. Silicon semiconductor technology allows very fast readout capability, which is necessary for collisions happening each 25 ns. 3.2.2 The Electromagnetic Calorimeter The electromagnetic calorimeter or ECAL surrounds the tracker, with an inner radius of about 1.3 m to an outer radius of about 1.7 m and covers a pseudorapidity range |η| < 3.0. Its function is to detect those particles that interact electromagnetically; in particular electrons/positrons and photons. It consists of a large collection of lead tungstate crystals (PbWO4 ) spread over the barrel and two endcaps. These kind of crystals scintillate, meaning they produce light proportional to the energy deposited in the crystal. This light is collected through Avalanche Photo Diodes (APDs) attached to one side of the crystal2 which eventually produce an electric signal in proportion to the incoming light, thus in proportion to the deposited energy. The crystals have a very short interaction length and can withstand very high doses of radiation. PbWO4 is an inorganic scintillator crystal with impurities added. When an energetic particle passes, electrons are elevated from valence to the conduction band. The electron is then free to move through the crystal after which it eventually recombines with a hole and produces light. The impurities are necessary to make the recombination more efficient, thus yielding very fast light outputs (See Fig. 3.6). In 25 ns, about 80% of the light produced from the energy deposit is emitted [32]. c G. F. Knoll, Radiation Figure 3.6: Schematic display of the PbWO4 band structure. detection and measurement. (2010) 2 APDs are used for the ECAL barrel section. For the ECAL endcap sections, vacuum phototriodes are used. 37 At the endcap on the inner side, the ECAL preshower system is installed on both sides. It has a finer granularity; silicon detector strips of 2 mm wide compared to the 3 cm crystals of the rest of the ECAL. One of its important design constraints is to be able to resolve nearby γ’s so that the Higgs boson decaying into two photons could be precisely reconstructed. This is studied using the π 0 → γγ process which is typically boosted and produces very small decay angles between the 2 photons, thus giving rise to a possible single hit in the 3 cm ECAL crystal. The smaller strips can distinguish between such hits. 3.2.3 The Hadronic Calorimeter The hadronic calorimeter or HCAL is situated in between the ECAL and the solenoidal magnet (next section) with inner radius of 1.77 m and outer radius of 2.95 m and covers a pseudorapidity range |η| < 3.0. It is designed to measure mainly the energy of hadrons, interacting strongly with the HCAL material. Due to the larger interaction length of hadrons, the HCAL must be larger than the ECAL. The HCAL is a sampling calorimeter3 , existing of 36 barrel and endcap wedges. It is made of alternating layers of dense brass (passive) absorber material and (active) plastic scintillator. The idea is that particles hit the absorber material and produce a cascade of shower particles. These deposit energy in the scintillation material again providing an electrical output proportional to the deposited energy. This is done through the use of optical fibres that conduct the light output into readout boxes where photodetectors amplify the signal. The shower has an electromagnetic and hadronic component (see Fig. 3.7). The hadronic component contains mostly π ± and π 0 that decay according to π ± → µ± + ν and π 0 → γγ. The hadronic and electromagnetic components of showers fluctuate grossly event by event. The HCAL thus provides information on the position of a particle, energy and arrival time. It is made as hermetic as possible, which allows indirect measurements of noninteracting particles. This is thanks to the fact that the sum of all transverse momenta should equal one, which follows from momentum conservation and the knowledge that pinitial = 0. The missing transverse momentum is then defined as pmiss = -Σi piT , where T T the sum i runs over all track momenta. 3 In a sampling calorimeter, the absorber material degrades the energy of the particle hitting the detector and this interaction produces a shower of particles. The absorber is alternated by layers of active material that produces the signals that could be read out. 38 Figure 3.7: A neutron hitting the absorber material and starting a particle shower, with both an electromagnetic and hadronic component. [33] 3.2.4 The Superconducting Solenoidal Magnet The superconducting magnet for CMS surrounds the tracker, the ECAL and the HCAL. It was designed to reach a 4-T magnetic field originating from a unique 4-layer solenoidal winding of NbTi conductor wires. The magnet is 6.4 m in diameter and 12.5 m in length, it weighs 220 metric tonnes. In order to reach the large magnetic field, very high currents must flow through the solenoidal wires. This is only possible if those wires are superconducting, which needs cooling of the magnet towards cryogenic temperatures of 4.5 K, done using liquid He. A 4-T magnetic field of this size is truly remarkable. Particles with very high momentum originating from the interaction point are only bent in strong magnetic fields and this allows CMS to pinpoint a particle’s momentum with high precision. Fig. 3.8 shows how different particles are bent under influence of the magnetic field, as well as the location of detection. The solenoid is surrounded by a 12.5 t iron return yoke (displayed in red in Fig 3.4 and 3.8), which contains and guides the magnetic field. It also functions as the main support of the entire detector. 39 Figure 3.8: Schematic view of CMS along the beam axis. Different particles movement and interaction locations are displayed. 3.2.5 The Muon System The detection of muons is of particular importance for CMS, as is clear from its name. CMS is designed in such a way that muon detection can be done with very high precision. Muons are not stopped by CMS’s calorimeters. The muon system is installed at the outer edge of the detector, where they are likely to be the only particles left to register a signal. A particle is measured by fitting a trajectory to a track from hits created in the four muon stations, which are situated outside the magnet coil and have return yoke plates in between them (denoted in red in Fig. 3.8). This combined with the tracker hits at the core of the detector allows a particle’s path to be precisely reconstructed, especially thanks to the high magnetic field to bend even the paths of very high-energy muons. Hits in the muon chambers originate from a combination of three types of gaseous particle detectors: (1) Muon drift tubes: These are rectangular shaped gaseous volumes (Ar/CO2 ) whose edges consist of cathode strips and through which an anode wire runs. A charged particle passing through the volume generates ionisation around the particle track where positive ions and negative electrons respectively move to the 40 cathode and the anode. Stacking groups of drift tubes on top of each other, allows from timing measurements of charge collection time to reconstruct the path of the muon. (2) Cathode strip chambers: These components are situated at the endcap disks of the detector. They consist of positively-charged anode wires who are perpendicularly crossed with negatively-charged cathode strips, all confined within a gas volume (Ar/CO2 /CF4 ). Due to the particular setup of those chambers and the electrical fields applied, electrons generated from the ionisation process trigger an avalanche-effect allowing precise measurements of the particle’s position. (3) Resistive plate chambers: These components are installed both on the barrel and endcap sections of CMS. They consist of two parallel plates, a positively charged anode and a negatively charged cathode, both made of high resistivity material, separated by a gas volume (C2 H2 F4 /i-C2 H10 ). They are also operated in avalanche mode and provide very fast detector timing. 3.2.6 Triggering and Data Acquisition When bunch crossings take place each 25 ns, about 1 million pp collisions take place every second4 . It is technically impossible to detect and read out all the data from each individual collision. Apart from technical reasons, plenty of those collisions are worthless as they might be low-energy collisions for instance, instead of highly energetic, and would be rejected from all interesting analyses in event selection. One clearly needs techniques to only select those events that are potentially interesting. Due to the tremendously high frequency of collisions, new events are happening, when the previous one did not even leave the detector yet. This is called pile-up and complicates the selection of interesting events, degrades measurements and sensitivity to new physics searches if analyses are not accoumpanied with sophisticated pile-up mitigation techniques. Everything that gets measured by CMS needs to pass a set of initial criteria before further, more strict selection criteria are imposed on events in the high level trigger and in specific analyses. This process is called triggering. The triggering happens in two stages in CMS. The first stage is the Level 1 (L1) trigger [34] and the second stage is the High Level Trigger (HLT) [35], [36]. The L1 trigger makes use of the calorimetry and muon systems. Its decision is based on the presence of particle candidates such as electrons, muons, photons and jets above certain ET or pT tresholds. It effectively 4 On average about 20-30 protons collide each bunch crossing. 41 reduces the 40 MHz event rate to 100 kHz. The HLT filters the rate even further to 100 Hz. The CMS DAQ/HLT system has the following functionality • Read-out after a Level-1 trigger accept. • Execute physics selection algorithms on the events that are read out. Accept those events that are physically most interesting. • Forward these events towards mass storage for offline computing. The HLT selection algorithms [36] use info from calorimeters, muon systems and pixel detectors. Then, the algorithms do primary vertex and full track reconstruction and btagging using secondary vertex identification (Fig. 3.5). It needs to be noted however, that to minimize the CPU required by the HLT, the algorithms only reconstruct the information partially. Full event reconstruction is done in offline computing. 42 Chapter 4 Modelling and Analysis Software This chapter will provide an overview of the software used in this thesis to perform the necessary physics analyses. A general outline of the CMS Software (CMSSW) Application Framework will be given as well as an introduction to Monte Carlo (MC) event generators. An overview of the different data-types that are relevant may be beneficial: • RAW: Detector data after it passed the L1 trigger and HLT selection criteria (see Sec. 3.2.6). • RECO: Reconstructed objects (tracks, vertices, jets, leptons, etc.) from all stages of reconstruction. • AOD: Subset of RECO data that is relevant for physics research in a convenient and compact format. • GEN: Generated Monte Carlo (MC) data. • SIM: Energy depositions of MC particles in the detector. • DIGI: SIM hits converted into digitised detector response. This is the equivalent of RAW data, but here with simulated MC data. 4.1 CMSSW Application Framework CMSSW [37] is a framework that provides the ability to deploy reconstruction and analysis software in order for physicists to be able to perform an analysis. It is built around an Event Data Model (EDM). An event in this context denotes a C++ object of 43 all RAW and reconstructed data for one particular collision. All the objects in such an event, are stored in a particular format called ROOT files. ROOT is an object-oriented framework aimed at solving the data analysis challenges of high-energy physics [38]. An important software toolkit used in CMSSW is GEANT4 (GEometry ANd Tracking) [39]. It is a platform for simulation of passage of particles through matter. It is used in a large variety of high-energy physics experiments. For CMS in particular, it allows MC data to be reconstructed as if it were real data. This means effects taking into account the geometry of the detector that can influence the reconstructed events are included. As was mentioned in Sec. 3.2.6, the 40 MHz event rate is reduced to about 100 Hz of RAW data events. This data is then processed towards RECO data objects. The latter dataset is still very large and contains irrelevant information for physics analyses and gets reduced to AOD data (Analysis Object Data). 4.2 Particle-flow Event Reconstruction Particle-Flow (PF) [40, 41] is an algorithm to reconstruct all stable particles in an event by combining the kinematical information from the different CMS detector parts. Photons deposit their energy in the ECAL and are thus reconstructed from ECAL clusters. The energy of electrons is determined from a combination of the electron momentum acquired by the tracker, the energy deposited in the ECAL cluster and the total energy originating from Bremsstrahlung photons that are compatible with the electron track. The energy of muons is determined from curvature measurements of its track, obtained from tracker and muon system information. The energy of charged hadrons is measured from a combination of tracker and ECAL and HCAL cluster information, when both values are found to be compatible1 And finally, the energy of neutral hadrons is obtained from ECAL and HCAL clusters. 4.3 Monte Carlo Event Generators MC generators are indispensable in hadron collider physics research. They comprise of a set of models to simulate actual events from a particular process. For the LHC, when 1 If the calorimeter response is to be too small, a cleaning procedure is performed to remove potential spurious or mis-reconstructed tracks. If, on the other hand, the calorimeter response is too large, the PF algorithm assigns the energy excess to a photon and possibly a neutral hadron. 44 two protons collide, a few very important things must be taken into account in order to produce a viable simulation of the interaction event. First of all, a hard scattering process must be implemented. This is the interaction between two partons (quarks or gluons) from each of the two protons. The hard scattering occurs at high momentum transfer and is described by the matrix element (ME). Furthermore, it can be computed in the perturbative regime thanks to the running of αS (Sec. 1.2.3). Afterwards, the process of parton showering (PS) will start. This means that quarks and gluons originating from the hard scattering are propagated and radiate additional quarks and gluons and finally undergo hadronization. Here, a transition from perturbative to non-perturbative regime occurs since every additional radiation lowers the amount of transferred momentum. The sequential steps in this process can be modelled using dedicated Monte Carlo programs. This means the modelling is done in a probabilistic way, which is an accurate description of what happens in nature. An introduction on two concrete and distinct examples of MC programs, POWHEG and PYTHIA, will be given in order to gain a better understanding of their functionality and use. 4.3.1 Powheg POWHEG [42], [43] is a Matrix Element (ME) Monte Carlo event generator. Given a particular process, e.g. pp → tt̄, it will generate all the possible Feynman diagrams that can give rise to the tt̄ state at a particular order in perturbation theory2 . It must be noted that this type of MC generators work in the perturbative regime. As was mentioned in Sec. 1.2.3, the strong coupling constant αS is a running constant, meaning it becomes smaller the larger the amount of transferred momentum gets. As POWHEG simulates the collision of quarks or gluons from a pp collision, there is no momentum lost to radiative processes yet, so the value of transferred momentum is very high. A scattering event in this regime is called a hard-scattering. POWHEG is able to calculate matrix elements up to Next-to-Leading-Order (NLO). This means that the generator also takes hard parton emission into account. Examples of LO and NLO diagrams are given in Figs. 4.1 and 4.2. They display examples of tt̄production diagrams, the f and f¯ are a fermion and an anti-fermion3 . Although the diagrams seem simple, NLO diagrams provide difficulties in theoretical calculations. 2 3 To view which processes can be computed with POWHEG, see [44]. The W-boson has multiple ways of decaying, e.g. W → lν or W → q q¯0 45 Figure 4.1: First order (LO) tt̄ production Figure 4.2: Example of a tt̄ production proprocess without additional radiations. cess with a gluon radiating from one of the initial quarks. The gluon gives rise to an extra vertex, causing the diagram to be at NLO. NLO diagrams provide not only positive, but also negative contributions to the crosssection. This makes it difficult to implement full NLO processes in MC generators. Another difficulty exists when the ME generator (e.g. POWHEG) is made to match with a PS code (e.g. PYTHIA). This is called merging, and will be explained in Sec. 4.3.2. 4.3.2 Pythia PYTHIA [45] is used to generate high-energy-physics events: sets of outgoing particles produced from the interactions between two incoming particles. It comprises a collection of physics models for the evolution of a hard scattering process to a complex multihadronical final state4 . The complexity stems also from the fact that many processes must be taken into account. First of all, the original Feynman-diagrams receive Strahlung-modifications, in which e → γe or q → gq type of processes result in additional final-state particles. Second, there are higher-order Feynman diagrams which sometimes combine to cancel divergences, resulting in perturbative techniques to become necessary. Typically this is only done up to one loop, which is referred to as an NLO-approximation. Third, due to confinement of quarks and gluons, no coloured particles can propagate freely. Therefore they must hadronize in colourless final states. This finally results in a collection of jets composed of photons, hadrons and leptons. 4 Note that PYTHIA usually does not simulate the interaction itself but is typically used for the PS. 46 Figure 4.3: An example of a PS process calculated in PYTHIA. Starting of with a transferred momentum of q12 , the parton radiates gluons losing momentum to q22 , q32 , etc. Once the parton lost enough momentum and passed a certain cut-off momentum, it is left to hadronise into a colourless final state. The entire problem can be summarised in two steps: first, the hard process is used as input to generate the corrections described in the previous paragraph and second it is left to hadronize. It is the task of PYTHIA to generate events (using MC techniques) such that its output (after simulation of the detector using GEANT4) is very similar to real data as measured by CMS. In typical data analysis, this is a very important aspect of the conducted study, since such generated events provide a tool for devising analysis strategies that can be used on real data. Furthermore they are of essential importance as they allow to model SM processes, make SM parameter predictions, and model backgrounds. It was alluded in Sec. 4.3.1 that difficulties exist when the ME generator functions at NLO. Since PYTHIA indeed produces a PS it is possible that a PS produced from a LO diagram, coincides with a diagram produced in NLO (for an example of a PS, see Fig. 4.3). The concept of merging [46] solves this by including algorithms in the MC generators to prevent double counting, so that the same diagrams aren’t added to the phase space5 . 4.3.3 Tuning of event-generators: Professor Monte-Carlo event-generators use various approximations. To ensure the validity of those approximations, tuning is done on its parameters, to find a match between the predicted set of events compared to what’s measured in data. With Eq. 1.8, we alluded that there was a perturbative and non-perturbative part to be taken into account to compute the cross section. Both the perturbative and the non-perturbative parts 5 The phase space is the mathematical space of all possible momenta of all the outgoing particles. 47 could be tuned. The UE has many parameters that can not be calculated theoretically, therefore tuning becomes necessary to have these parameters implemented correctly. PS programs also have some tunable, free parameters such as αSISR . There’s multiple ways for doing so: either one does it manually which requires great skill and luck, or one does it using dedicated programs optimised for tuning MC-generators in the most efficient way. Professor [47] is a program dedicated to tune model parameters of MC event-generators to experimental data by parametrising the per-bin generator response to parameter variations and numerically optimising the parametrised behaviour. Typical eventgenerators have O(10-30) parameters that are relevant for collider physics simulations. It goes without saying that sampling in a multi-dimensional hyperspace is computationally very costly. If P is the number of parameters to be tuned, then for each variation of the P -element parameter vector p = (p1 , ..., pP ) a new event-generator computation would be necessary. The idea here, would be to define a goodness of fit (GoF) function between a limited set of MC data and the actual real data, and then to minimise that function corresponding to an optimal parameter vector pbest . A more elaborate discussion on the functioning of Professor can be found in appendix A. 48 Chapter 5 Analysis Part I: Search for Scalar Top Quark Partners 5.1 Introduction The first part of the analyses done in this thesis revolves around a SUSY search for ∗ stop √quark (t̃t̃ ) production conducted in the opposite sign dilepton channel at the LHC at s = 13 TeV pp collisions. We look for the signal in which both W bosons decay leptonically (i.e. to ee, µµ or eµ), hence the name: dilepton channel. It is depicted in Fig. 5.1. q̄ b t W t̃ g t̃ + l+ νl χ̃01 χ̃01 ∗ W− t̄ l− ν̄l b̄ q Figure 5.1: A hypothetical t̃t̃∗ pair is produced after a pp collision at the LHC. When assuming R-parity conservation, the stop quarks are produced in pairs and decay to a top quark and a neutralino (the latter is the LSP, invisible in our detector). The W bosons, originating from top quarks, both decay leptonically. 49 Figure 5.2: Branching fractions of top quark decay. Each time the decays of the two W bosons are denoted. Full hadronic means both the W-bosons decay into two quarks and thus produce jets. Note that the decay of the unstable τ -lepton as τ → e/µ + νe/µ has not been included. The branching fractions of the top quark decay are given in Fig. 5.2. It is clear that the dilepton channel accounts for only a small fraction of top pair decays (9%). The full hadronic channel for example, accounts for a branching of 46%. There are plenty of processes that have large jet activity making it difficult to discriminate signal from background. The dilepton channel on the other hand produces electrons or muons1 . Those are very easily detectable with the CMS ECAL (Sec. 3.2.2) and Muon systems (Sec. 3.2.5). In SUSY-searches, there is another very important difference compared to regular SMsearches. There are no a priori predictions on the masses of the supersymmetric particles. This means that the energy available for the top quarks depends on the masses of the stop t̃ and neutralino χ̃01 , which aren’t known. One therefore always has to make a prediction on the mass difference ∆(mt̃ , mχ̃01 ). Depending on this mass difference, more or less energy will be available for top quark production and this will have a consequence on how energetic its decay products are. A particularly difficult signal region to search for is the compressed signal, for which ∆(mt̃ , mχ̃01 ) is small. This means the energy available to the top quark is only 100 1 τ + The tau-lepton dominantly decays into electrons or muons through τ − → ντ + e− /µ− + ν̄e /ν̄µ and → ν̄τ + e+ /µ+ + νe /νµ . 50 GeV/c2 compared to the 173.3 GeV/c2 needed to produce it on-shell. The top quark will therefore be off-shell and produces very soft b quarks, leptons and neutrinos. Due to the presence of two LSPs and two neutrinos from W decay, the signal has a large amount of ETmiss , which will typically contain more ETmiss than regular SM processes which can be exploited by constructing a particular type of variable, MT 2 (ll). This variable will provide a way of making a distinction between the signal region and the background region. 5.2 MT 2(ll) We now introduce an important variable for this study called MT 2 (ll) [48] offering the possibility of selecting a signal region. The variable is defined by MT2 2 (ll) = min miss miss pmiss l1 +pl2 =ET miss 2 l2 miss max MT2 pl1 , p , M p , p T l1 T T l2 (5.1) miss = ETmiss means and is constructed as listed below. Note that the condition pmiss l1 + pl2 miss miss miss is defined in Sec. = |pmiss that |pmiss T |. The variable pT l1 | + |pl2 | must equal ET 3.2.3. and pmiss (1) The ETmiss is partitioned into two hypothetical neutrinos with pmiss l2 . l1 (2) The transverse mass MT is computed for each lepton with one of the hypothetical neutrinos. (3) The maximum of those two values is stored, the other is discarded. (4) This is repeated for every partition of ETmiss and then the minimum is taken. That value is the event’s MT 2 (ll). Note that if in the computation for MT 2 (ll) a maximum of 0 GeV/c2 is found for a given partition of ETmiss , MT 2 (ll) = 0 GeV/c2 will be kept. For a particular set of events this will always be the case when the ETmiss lies within the smaller angle subtended by the pT of the two leptons. For those events, a partition will be encountered with the two neutrinos aligned with the pT of the leptons causing to yield a value of MT = 0 GeV/c2 . Therefore a peak in the MT 2 (ll) spectrum at 0 GeV/c2 will be typically seen. For dileptonic tt̄ and W W events (some of our SM backgrounds), it can be shown [49] that MT 2 (ll) has the same property for leptonic and single W events, namely that the 51 distribution has an endpoint at the mass of the W boson. However, two neutralinos as illustrated in the process in Fig. 5.1 give rise to additional ETmiss . This will cause the partitioning of ETmiss in Eq. 5.1 to create more energetic neutrinos than they actually are, thus causing MT 2 (ll) to possibly have a higher value than for example a regular tt̄ process. For top squark pair events, the distribution in MT 2 (ll) will depend on the value of ∆(mt̃ , mχ̃01 ). For large top squark masses, the neutralinos are boosted and this additional ETmiss will push the MT 2 (ll) distribution past the value of MW . If the mass difference between the top squark and top quark is small, the neutralinos will be produced relatively at rest and will cause the MT 2 (ll) distribution to look like the one found from SM tt̄ events. 5.3 Analysis strategy This section will provide an overview of the analysis strategy taken to construct an optimal selection2 . The selection requires plepton > 20 GeV/c, |η| < 2.4, miniRelIso < 0.2 T (see Sec. 5.5), dxy < 0.05, dz < 0.1 for the leptons, and pjet T > 30 GeV/c, |η| < 2.4 for the jets. An overview of the Feynman diagrams of the most important MC generated backgrounds (processes that give rise to the same signature as the signal) is given on page 563 . It is easily seen from the Feynman diagram in Fig. 5.1 that we will have to put requirements on Njets and Nb-jets . Therefore, at least two jets are required of which at least one is b-tagged. The most dominant background consists of Drell-Yan (DY) production: two leptons with opposite signs originate from Z boson decay. It needs to be drastically reduced by applying specific cuts. A regular DY process does not have any ETmiss . It is possible however, that a gluon radiates from the initial incoming quark pair. This produces a jet opposite to the lepton pair, as depicted in Fig. 5.3. If the pT of this jet is over- or undermeasured in the detector, fake ETmiss is reconstructed. 2 The selection should leave the number of signal events as untouched as possible, but should reduce backgrounds as much as possible. 3 The backgrounds from QCD- and tt̄ + X production (with X a W or a H boson) are not given there but are also present in the analysis. They are very similar to process (3), but with a W - or H-boson radiated from the top quark instead of a Z 0 boson. 52 Since we require at least two jets, it is possible that two gluons are radiated from the incoming quark pair, and thus their energies get overmeasured4 . Depending on overor undermeasurements, the ETmiss is opposite to the jet or aligned with the jet. By requiring cos(φETmiss − φjet1,2 ) < cos(0.25), events with jets aligned and opposite to the ETmiss will be omitted from the analysis. A second pair of cuts that effectively reduces DY is cutting directly on ETmiss and the ETmiss -significance (see Ref. [50]). The latter is defined by E miss ETmiss -significance = √T HT (5.2) √ with HT equal to the sum of the pT of all reconstructed jets. HT is a measure for the resolution of the reconstructed ETmiss . The chance that invisible particles contributed to ETmiss is significantly higher if the value of ETmiss is larger than its resolution. Requiring the event to have an ETmiss -significance larger than a certain treshold reduces the DY background as the ETmiss will be of the order of the resolution. This won’t affect our signal as it should have a large ETmiss due to the neutralinos. Figure 5.3: A jet radiating in the opposite direction of the lepton pair can get over- or undermeasured and account for fake ETmiss . If the jet energy is overmeasured, we have ∆(φETmiss , φjet ) ≈ π. If it is undermeasured, we have ∆(φETmiss , φjet ) ≈ 0 4 Jet mismeasurements occur from noise effects, varying detector response, pile-up contributions, etc. 53 To veto events in which the leptons originate from Z decay, a Z-veto cut is applied. Concretely, we require |mll − mZ | > 15 GeV/c2 for same flavour (SF) leptons with mll equal to the invariant mass constructed from both leptons. As the prompt leptons coming from W decay are highly energetic, a reconstructed invariant mass cut of mll > 20 GeV/c2 is required. All the applied cuts (including some that weren’t mentioned, but are trivial) and their effect on the backgrounds and signal are given in Table 5.1 in a cutflow diagram. Notice how all backgrounds are efficiently reduced by the MT 2 (ll)-cut except for tt̄ + jets, tt̄ + Z and tt̄ + X. The latter two are easily explained as they are irreducible backgrounds. This can be understood by looking at the tt̄ + Z diagram (5) on page 56. The diagrams for tt̄ + X are not given, but contain tt̄ + W and tt̄ + H production. If the Z boson or H boson decay gives rise to two neutrinos, then these backgrounds mimic our signal precisely causing them to end up in the high MT 2 (ll)-tails. The presence of the tt̄ + jets background is not easily explained and requires more attention (Sec. 5.4). 54 Cut Two well reconstructed leptons Opposite Sign (e± e∓ , µ± µ∓ , e± µ∓ ) mll > 20 GeV/c2 |mll − mZ | > 15 GeV/c2 for SF ≥ 2 jets ≥ 1 b-tags ETmiss > 80 GeV √ ETmiss / HT > 5 cos(φETmiss − φjet1,2 ) < cos(0.25) MT 2 (ll) > 140 GeV/c2 Cut Two well reconstructed leptons Opposite Sign (e± e∓ , µ± µ∓ , e± µ∓ ) mll > 20 GeV/c2 |mll − mZ | > 15 GeV/c2 for SF ≥ 2 jets ≥ 1 b-tags ETmiss > 80 GeV √ ETmiss / HT > 5 cos(φETmiss − φjet1,2 ) < cos(0.25) MT 2 (ll) > 140 GeV/c2 DY 13333791.3 13281921.7 13189194.4 1185896.0 52475.3 5048.1 84.2 61.0 46.9 0.4 W + jets 5554.2 3679.0 3357.3 3140.3 928.4 86.7 16.8 14.9 13.3 0.1 tt̄ + jets 218650.8 214128.7 208590.9 185391.0 137452.0 108792.2 44724.8 41028.4 38318.5 2.9 Diboson 39734.0 38301.2 37437.8 18819.1 2475.6 206.7 54.7 50.4 47.8 0.3 tt̄ + Z 496.2 450.0 439.6 222.1 204.8 155.2 76.9 63.3 58.2 1.6 Triboson 142.7 132.1 131.1 48.1 28.4 5.6 2.7 2.4 2.3 0.0 tt̄ + X 884.5 687.6 673.3 436.5 399.7 315.3 163.6 130.1 119.1 1.1 QCD 255025.4 233498.5 37467.9 36338.5 1860.0 67.3 0.0 0.0 0.0 0.0 Single t 20550.0 19900.4 19413.5 17378.8 6497.2 4567.2 1940.3 1814.0 1699.0 0.6 Signal 296.8 293.5 291.1 261.1 206.0 168.7 149.8 146.3 136.4 26.1 Table 5.1: Cutflow-diagram of the applied cuts in the selection for the OS dilepton t̃t̃∗ -search. The numbers represent the number of MC events produced. The signal is found on the bottom right, the precise value of ∆(mt̃ , mχ̃01 ) is not given here as it is of c Schöfbeck R. less importance. 55 (1) Drell-Yan production (2) tt̄ + jets production q̄ b t g t̄ l+ W+ νl W− l− ν̄l b̄ q (3) Single top production (4) W + jets production q0 b g b̄ l W ν q̄ (5) tt̄ + Z production (6) Diboson production q0 t νl W Z0 t + ν̄l g l+ l− νl t̄ W q̄ − ν̄l (7) Triboson production q 0 νl W+ W+ l+ q̄ Z0 q W q̄ − l 1 − ν̄l 56 5.4 MT 2(ll)-tail events study for the tt̄ + jets background The large amount of tt̄ + jets events in the tails of MT 2 (ll) is worrying. This is due to the fact that following from Eq. 5.1 no tt̄ events should end up in the high tails since the only ETmiss can originate from the two neutrinos. We define the high-tail region as MT 2 (ll) > 140 GeV/c2 . For this region an explicit check will be done to find out what the kinematics of tt̄ + jets events exactly look like in full detail. The tt̄ + jets background in function of MT 2 (ll) is displayed in Fig. 5.4. Figure 5.4: Number of tt̄ + jets events in function of MT 2 (ll). The expected cutoff around 80 GeV/c2 is visible (notice the log scale), however there are still events surviving into the high-tail region (> 140 GeV/c2 ). A huge advantage in this study, is that in MC we have access to the generated particle’s kinematical information (GEN info), which is not the case in data. We can thus explicitly see if something went wrong in the process of reconstructing the particles. In reconstruction (RECO), the interactions in the detector are also accounted for, which inevitably causes incorrect reconstructions of particles. We are particularly interested in the leptons that were selected as the prompt leptons to see whether they are in fact the decay products of the W bosons. If not, we are selecting the wrong leptons to compute the event’s MT 2 (ll) and that could explain why we see events ending up in the high tails. 57 GEN - Level RECO - Level Access to simulated kinematical information. → Monte Carlo ⇐⇒ Access to reconstructed kinematical information. → Monte Carlo → Data In the tail-region (MT 2 (ll) > 140 GeV/c2 ), 37 tt̄ + jets events are counted. For those events, the GEN-information is matched to the RECO-information, of which an example is displayed in Table 5.2. This means that for each of the two RECO leptons, the corresponding GEN lepton is identified. For the GEN leptons, the PdgIds5 of the mother particles are checked (the one from which the lepton decayed). Once these are found and one or more are not a W boson, the mother particle (e.g. a b quark) is again RECO Leptons φ Lepton pT GeV/c 90.660 0.294 (1) µ 63.470 0.314 (2) e Lepton (1) µ (2) e Quark b GEN Leptons pT GeV/c φ 89.290 37.590 0.294 0.314 GEN Quarks pT GeV/c φ 131.86 0.340 η 0.937 1.637 η RECO ∆R : l(1) RECO ∆R : l(2) Mother 0.937 1.637 0.000 0.701 0.699 0.002 W b η RECO ∆R : l(1) RECO ∆R : l(2) 1.647 0.712 0.029 Table 5.2: Example of how the GEN-RECO matching for a high-tail MT 2 (ll) tt̄ + jets event is done. First p the RECO leptons are matched to the GEN leptons, based on demanding ∆R = ∆φ2 + ∆η 2 ≈ 0. Then the PdgId of the GEN leptons is checked to what the mother particle of the leptons is. If it is found that this is a b quark and ∆R(lRECO |b) < 0.06, the b quark is said to be matched to the lepton, indicating that the lepton originated from a b-decay. The matched leptons and b quark are shown by the interconnecting lines. 5 The PdgIds are the conventional numbering IDs for all elementary particles set by Particle Data Group [51]. 58 matched to its corresponding lepton based on the fact that their separation should not be larger than ∆R = 0.06. Of the 37 high-tail events, 29 out of 37 events have at least one selected lepton decayed from a b quark and not a W boson and 8 out of 37 events are originating from W decay. This means that ≈80% of MT 2 (ll) tail events are fakes; namely wrongly selected leptons or fakes. 5.5 Reduction of the tt̄ + jets background In order to reduce the fakes in the high tails of MT 2 (ll), an efficient way of making a distinction between leptons originating from a W boson compared to a b quark decay must be found. A first suggestion is to use the Reliso0X variable. It is constructed for every selected lepton by drawing a circle of ∆R = 0.X in the angle space with center around the lepton’s track and by computing P RelIso0X = i∈all RECO particles plep T piT (5.3) where i only runs over pT s that are inside the predefined circle as shown in Fig. 5.5 (a). Since leptons that originate from b quarks will typically have lots of jet activity in their vicinity, a RelIso0X < Y cut can reduce the contribution of those leptons. A rather small cone (also referred to as MiniRelIso) does not help reducing the fakes in the tails. The idea is that a lepton can be accidentally6 radiated away from the jet activity (see Fig. 5.5 (b)) and a small cone cannot catch enough jet tracks for the event to fail the cut. If one however adds a Reliso04 < 0.12 cut, where a larger cone size of 0.4 is chosen, approximately 80% of fakes in the tails can be reduced [52]. The problem with this approach, however, is that this also touches prompt leptons (those who originate from W decay). It seems that the RelIso04 < 0.12 cut is not a good choice after all. A new variable, that leaves the prompt leptons untouched, but removes specifically those originating from semileptonic b quark decay, must be constructed. 6 This might seem to be unlikely, but we are looking at events in high tails of MT 2 (ll), so we expect things to be unlikely! 59 (a) (b) (c) Figure 5.5: Display of the RelIso0X approach with conesize 0.X for (a) a lepton near jet activity (usual case), (b) a lepton radiated away from jet activity with conesize 0.X and (c) a lepton ejected away from jet activity with a conesize 0.Y > 0.X. This new approach is to construct the MultiIsoV2 variable, defined as MultiIsoV2 = MiniRelIso < A && (ptRatioV2 > B k ptRelV2 > C) . (5.4) It contains three components, explained below: • MiniRelIso: This is the same principle as RelIso, but corrected for boosts. The cone size gets a pT dependence: the higher the pT of the lepton is, the smaller the cone size necessary to catch jet activity as the jet will have a much smaller opening angle. Jet • ptRatioV2: It is defined as plep T /pT , where the closest jet to the lepton is considered. If the lepton has a significantly larger pT than the jet, chances are small it originated from the b quark. • ptRelV2: It is defined as k~pTlep − p~Jet T k and is displayed in Fig. 5.6a. The larger this value, the further the lepton will be separated from the jet, so the smaller the chances are it originated from the b quark. 60 (a) Display of how ptRelV2 is computed. (b) Approximately 80% of fake high tail tt̄ + jets events are killed by the MultiIsoV2 cut. Figure 5.6 The MultiIsoV2 cut acting on the tt̄ + jets background is displayed in Fig. 5.7. Here, A, B, C were taken to be respectively 0.09, 0.84, 7.2. It’s clear how the low MT 2 (ll) region gets reduced by about 15%, but from about 110 GeV/c2 onwards, a clear reduction of tail events becomes visible. Events / 10 GeV/c2 MultiIsoV2 = miniRelIso(l ,l ) < 0.09 && (jetPtRatiov2(l ,l ) > 0.84 || jetPtRelv2(l ,l ) > 7.2) 1 2 1 2 MultiIsoV2 thus effectively reduces fakes in the high MT 2 (ll) region (≈80%, see Fig. 5.6b). Over the entire MT 2 (ll) spectrum, the tt̄ + jets background is reduced by approximately 48% and the signal only loses about 4% [52]. 1 2 tt + Jets (lep) 103 tt + Jets (lep) + MultiIsoV2 2 10 10 1 10− 1 10− 2 1.20 1 0.8 0.6 0.4 0.2 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 MT2(ll) Figure 5.7: Effect of the MultiIsoV2 cut on the tt̄ + jets background in the MT 2 (ll) distribution before the cut (cyan) and after the cut (red). 61 5.6 Summary and Discussion In this section, an analysis is presented within the scope of an opposite sign SUSYsearch for stop quarks, with the production process depicted in Fig. 5.1. An overview of the analysis strategy is given along with the most important cuts that need to be applied in order to reduce background processes as much as possible. Their effect can be seen in the cutflow-diagram presented in Table 5.1. The various backgrounds that have to be taken into account are presented with their corresponding Feynman diagrams on page 56. It is noted that the tt̄ + jets background requires extra attention. This can be seen from Fig. 5.4 and also from the cutflow-diagram in Table 5.1 where tt̄ + jets events appear in the high tails of MT 2 (ll) (> 140 GeV/c2 ). As is explained in Sec. 5.2, events originating from a SM tt̄ process shouldn’t be found for high values of MT 2 (ll). To find which selected leptons are present in the high-tails, GEN information is compared to RECO information to identify the mother particles of the selected leptons. An example of this is given in Table 5.2. It is found that 29 out of 37 events in the tails have leptons originating from a b quark instead of a W boson. This means that for those events, the wrong leptons are selected, and should be categorized as fakes. Two approaches are suggested to reduce the fakes in the tails. The first, adding a RelIso04 < 0.12 cut, does not turn out to be efficient as also prompt leptons are touched by it. This is presented in [52]. Therefore, a MultiIsoV2 cut is added instead as defined in Eq. 5.4. This cut seems to be very efficient in reducing the tails and leaves the signal untouched. The tt̄ + jets background is reduced by 48% in total with a loss of 80% in the tails and the signal is only reduced by 4% in the tails. This can also be seen from the pie chart in Fig. 5.6b (b) and Fig. 5.7. 62 Chapter 6 Analysis Part II: Parton Shower Tuning in Modelling Top Quark Pairs 6.1 Introduction MC event generators in high energy physics employ various physics models to describe a large variety of different types of processes [53]. Their results are used to devise search strategies or background estimates of physical processes to allow theoretical interpretations of experimental results. MC generators should provide a good description of available data, providing confidence in the theoretical models underlying the generator performance. Due to the large amount of data available from the LHC and previous experiments, experimental uncertainties have become more precise. This allows detailed studies on the validity of MC simulations and tuning (Sec. 4.3.3) efforts pave the way towards new and more precise predictions. The free parameters of these models need to be tuned from distributions of suitable observables extracted from experimental measurements. It has been known that the Njets observable as depicted in Fig. 6.1 is not well simulated by PYTHIA8, causing inaccurate MC outputs. Overestimation of MC events (generated with POWHEG+PYTHIA8) with respect to data becomes visible for larger Njets bins. This has also been observed with 13 TeV data (in the dilepton channel [54] and the single lepton channel [55]) as well as with MADGRAPH5 aMC@NLO+PYTHIA8 in both FxFx and MLM configurations1 . Among others, two main causes for this are identified: (1) The 1 Those are two merging schemes used in MADGRAPH5 aMC@NLO for NLO and LO cases respectively, 63 matching scheme between POWHEG and PYTHIA8, or (2) some of the PYTHIA8 parameters are not optimal, where a combination of both issues might be a possibility too. In this particular study, an effort is made to explore whether solutions can be found using dedicated tools to perform a tune on one of the parameters in PYTHIA8. To tackle this issue, αSISR , a PYTHIA8 parameter that controls Initial State Radiation (ISR) effects in the parton shower process could be tuned [56]. The tuning tool used is Professor [47], a program for tuning model parameters of MC event generators to experimental data by parametrising the per-bin generator response to parameter variations and numerically optimising parametrised behaviour. More information on the functioning of Professor can be found in Appendix A. The POWHEG event generator provides a matrix element calculation of the pp → tt̄ up to an additional jet (not present in the LO calculation) matched to parton shower MC programs (in our case PYTHIA8). 19.7 fb−1 (8 TeV) 1/σvis dσvis /dNjets CMS b b 10−1 b Data POWHEG+PYTHIA8 b b Theory/Data 10−2 b 1.4 1.2 1 0.8 0.6 2 3 4 5 6 Njets [ p T > 30 GeV] Figure 6.1: Normalised √ tt̄ Cross-section in the dilepton channel in function of Njets (pT > 30 GeV) for s = 8 TeV (black) and the POWHEG+PYTHIA8 MC generator output (red). see Sec. 4.3.2 for a brief discussion on merging. 64 6.2 Tuning of αSISR In this study, we tune the SpaceShower:alphaSvalue parameter2 of PYTHIA8, here denoted by αSISR . 13 MC runs are generated for different values of αSISR . The used values are listed in Table 6.1. Run 0 1 2 3 4 5 6 αSISR 0.100 0.105 0.110 0.115 0.120 0.125 0.130 Run 7 8 9 10 11 12 αSISR 0.135 0.140 0.145 0.020 0.050 0.080 Table 6.1: Tabular overview of 13 MC runs for different values of αSISR . For runs 0-9, αSISR ∈ [0.100, 0.145] is chosen, since it is expected that αSISR would lie in between these bounds. For runs 10-12 however, smaller values are selected to make sure the tuning would not navigate towards them, which would clearly be a wrong result. Briefly, the Professor method goes as follows. A MC response function M Cb (p) is constructed for every observable’s bin b. Afterwards, for every point p in parameter space, a χ2 function is constructed according to χ2 (p) = X X (M Cb (p) − Rb ) O b∈O ∆2b , (6.1) where O denotes the observable, Rb the data at bin b, and ∆2b the total uncertainty originating from the quadrature sum of MC-uncertainty and data-uncertainty. Note that this formula corresponds to Eq. A.7 from Chapter A, with the weights wi chosen equal to 1. For every parameter point p, the χ2 is then interpolated and eventually ISR minimised. The value of p corresponding to a minimal χ2 , pbest (here αS,best ) is the final tune. This study selects four observables to which a tune can be performed. They are Njets for pT > 30, 60 and 100 GeV/c and subleading additional3 jet pT . Varying the values of αSISR for different MC-runs yields a different output in each observable bin. The outliers 2 For a full list of the PYTHIA8 parameters, refer to [57]. The subleading additional jet denotes the fourth most energetic jet in the dilepton channel. The two b-jets and one ISR-jet are produced in POWHEG (at NLO) and are not sensitive to αSISR . 3 65 of the αSISR variations are called envelopes and are displayed in Figs. 6.2 (a), (b), (c) and (d). Of the four observables, three are used to conduct three tuning efforts. They are listed below. 1 Njets [pT > 30 GeV] 2 Njets [pT > 60 GeV] 3 Njets [pT > 30 GeV] and Subleading additional jet pT Table 6.2: Different observables used for tuning. They are also displayed in Fig. 6.2. For each of the three tuning efforts, Professor performs a tune towards the observables mentioned in Table 6.2. In case of the Njets -observables, only bins with Njets > 3 are considered in the tuning process, where jets predominantly originate from the parton shower and hence these bins are sensitive to αSISR . Note that the distribution Njets (pT > 100 GeV) displayed in Fig. 6.2 (c) is not used in either of the combinations given in Table 6.2. The CMS analysis code used is from TOP-12-041 [58] along with the Rivet routine CMS 2015 I1397174. 66 19.7 fb−1 (8 TeV) Envelope Data b b b 10−1 19.7 fb−1 (8 TeV) CMS 1/σvis dσvis /dNjets 1/σvis dσvis /dNjets CMS b b b Envelope Data b 10−1 b b b b 10−2 b b 1.4 Theory/Data Theory/Data 10−2 1.2 1 0.8 0.6 2 3 4 1.4 1.2 1 0.8 0.6 5 6 Njets [ p T > 30 GeV] 0 19.7 fb−1 (8 TeV) CMS 1 2 1/σvis dσvis /dNjets b b b Envelope Data b 10−1 b 10−2 19.7 fb−1 (8 TeV) b 10−2 b b Envelope Data b 10−3 b 10−4 b b 1.4 Theory/Data Theory/Data 4 5 Njets [ p T > 60 GeV] (b) CMS 1/σvis dσvis /dp T [GeV−1 ] (a) 3 1.2 1 0.8 0.6 0 1 2 3 4 Njets [ p T > 100 GeV] 10−5 1.4 1.2 1 0.8 0.6 50 (c) 100 150 200 250 300 350 400 Subleading additional jet p T [GeV] (d) Figure 6.2: Envelope-plots for the four different possible observables that can be used for tuning. The blue bands denote how MC variations in αSISR from Table 6.1 overlap with the data. The yellow bands display the data uncertainties. 67 6.3 Estimation of uncertainty on αISR S Uncertainties on the tuned value of αSISR are estimated in two ways. The first method (1) is to consider the data uncertainties. For each observable combination, two additional datasets are constructed by varying the central data up with the upper error, and varying it down with the lower error. After this, the three datasets are used to perform three tunes. The resulting tune using the central dataset is the central tune result. The ones resulting from the upwards scaled and downwards scaled datasets are the respective upper and lower bounds of the central tune results4 . The second method (2) is to construct two eigentunes for each tune result5 . The parameter-space used here is one-dimensional and thus the eigentunes are found by varying the tuned result up and down with the value of the χ2 (this is the standard Professor -prescription). The values of αSISR corresponding to those variations are then its upper and lower bounds. The results can eventually be displayed for the studied observables alongside the central tune. A more thorough explanation on the construction of eigentunes is given in Appendix A.3. 6.4 Results Result αSISR (central) 1 0.115 2 0.104 3 0.113 Uncertainties (1) +0.021 -0.019 +0.014 -0.012 +0.029 -0.021 Percentages 18.635% 19.048% 13.114% 11.555% 25.402% 18.296% Table 6.3: of the tuning results taking the three different tunings as listed in Table 6.2 as input. The result of each tuning is summarised in Table 6.3. For each tuning one, the central tuning result is given along with its uncertainties according to method (1) described in Sec. 6.3, both in absolute values as percentages. The Professor output from taking the eigentunes into account, according to method (2), are omitted for this set of tuning 4 Note that for the upwards and downwards scaled datasets, the uncertainties in data are kept constant. 5 This can be done using the –eigentunes option in the prof-tune command. 68 efforts. The reason for this is that the tuning is done to only one or two distributions at most for typically a low amount of degrees of freedom. Therefore the resulting χ2 is small and varying up the χ2 by itself to acquire the errors yields unrealistically low uncertainties on the tuned value of αSISR . The results are displayed in Figs. 6.4, 6.5 and 6.6. The resulting tunes are displayed in Fig 6.3, along with the ATLAS ATTBAR-POWHEG tune [59], the PYTHIA8 (CUETP8M1) and PYTHIA6 (CTEQ5M) defaults. Due to the large data-uncertainties on the subleading additional jet pT observable, it isn’t taken into account for the further remaining part of the study. The optimal tuning result is chosen to be combination 1 in which the tuning towards the Njets (pT > 30 GeV) distribution was performed as its result lies well within experimental data-uncertainties independent of the jet pT treshold. The value of αSISR = 0.115+0.021 −0.019 is thus considered to be the optimal tune. CMS Tune 1 30 021 0.115−+00..019 CMS Tune 2 60 014 0.104−+00..012 CMS Tune 3 30 029 0.113−+00..021 Njet Njet Njet + sub leading add. jet pT ATLAS ATTBAR-POWHEG ATL-PHYS-PUB-2015-007 (7 TeV data) 0.121 CUETP8M1 0.137 CTEQ5M 0.127 0.050 0.100 0.150 αSISR Figure 6.3: Summary of tune results and their uncertainties. Two alternative tunes are presented in addition to the previously described one. The input distribution(s) for each tune is indicated below the label of each tune. The αSISR values in the ATLAS ATTBAR-POWHEG tune for PYTHIA8, the CMS CUETP8M1 tune for PYTHIA8 (CMS default in Run II), and the CMS Z2* PYTHIA6 tune (CMS default in Run I), using the CTEQ5M PDF set, are also displayed. 69 1/σvis dσvis /dNjets 1/σvis dσvis /dNjets b b 10−1 b CMS data 19.7fb−1 (8 TeV) arXiv:1510.03072 Data Uncertainty b b b b CMS data 19.7fb−1 (8 TeV) arXiv:1510.03072 Data Uncertainty b dummy b POWHEG+PYTHIA8 CUETP8M1 Theory/Data 1.4 1.2 1 0.8 0.6 2 3 4 b POWHEG+PYTHIA8 CUETP8M1 10−2 αISR S = 0.115 αISR S = 0.096 ISR αS = 0.137 10−3 Theory/Data 10−1 dummy 10−2 b b αISR S = 0.115 αISR S = 0.096 ISR αS = 0.137 b 1.4 1.2 1 0.8 0.6 5 6 Njets [ p T > 30 GeV] 0 1 2 b b b 10−1 CMS data 19.7fb−1 (8 TeV) arXiv:1510.03072 Data Uncertainty b POWHEG+PYTHIA8 CUETP8M1 αISR S = 0.115 αISR S = 0.096 αISR S = 0.137 b dummy 10−1 POWHEG+PYTHIA8 CUETP8M1 b 10−2 CMS data 19.7fb−1 (8 TeV) arXiv:1510.03072 Data Uncertainty αISR S = 0.115 αISR S = 0.096 αISR S = 0.137 b b 10−3 b b 10−3 1.4 1.2 1 0.8 0.6 0 1 10−4 b Theory/Data Theory/Data b dummy 10−2 4 5 Njets [ p T > 60 GeV] (b) 1/σvis dσvis /dp T [GeV−1 ] 1/σvis dσvis /dNjets (a) 3 1 2 3 4 Njets [ p T > 100 GeV] 10−5 1.4 1.2 1 0.8 0.6 50 (c) 100 150 200 250 300 350 400 Subleading additional jet p T [GeV] (d) Figure 6.4: Results of αSISR tuning 1. Jet multiplicity distribution, Njets (with pjet T > 30 GeV), after tuning αSISR with the Njets > 3 bins (where jets predominantly originate from the parton shower) is used as input to Professor [47]. The unfolded CMS data are shown with total error bars. In each plot, the calculated distribution assuming the tuned αSISR is shown (solid line). The calculated distributions with the lower bound (dashed line) and the upper bound (dot-dashed line) of the tuned αSISR are also displayed. Beneath each plot is shown the ratio of theory predictions to data. The yellow bands indicate the total data uncertainty. 70 1/σvis dσvis /dNjets 1/σvis dσvis /dNjets b b 10−1 b CMS data 19.7fb−1 (8 TeV) arXiv:1510.03072 Data Uncertainty b b b b CMS data 19.7fb−1 (8 TeV) arXiv:1510.03072 Data Uncertainty b dummy b POWHEG+PYTHIA8 CUETP8M1 Theory/Data 1.4 1.2 1 0.8 0.6 2 3 4 b POWHEG+PYTHIA8 CUETP8M1 10−2 αISR S = 0.104 αISR S = 0.092 ISR αS = 0.117 10−3 Theory/Data 10−1 dummy 10−2 b b αISR S = 0.104 αISR S = 0.092 ISR αS = 0.117 b 1.4 1.2 1 0.8 0.6 5 6 Njets [ p T > 30 GeV] 0 1 2 b b b 10−1 CMS data 19.7fb−1 (8 TeV) arXiv:1510.03072 Data Uncertainty b POWHEG+PYTHIA8 CUETP8M1 αISR S = 0.104 αISR S = 0.092 αISR S = 0.117 b dummy 10−1 POWHEG+PYTHIA8 CUETP8M1 b 10−2 CMS data 19.7fb−1 (8 TeV) arXiv:1510.03072 Data Uncertainty αISR S = 0.104 αISR S = 0.092 αISR S = 0.117 b b 10−3 b b 10−3 1.4 1.2 1 0.8 0.6 0 1 10−4 b Theory/Data Theory/Data b dummy 10−2 4 5 Njets [ p T > 60 GeV] (b) 1/σvis dσvis /dp T [GeV−1 ] 1/σvis dσvis /dNjets (a) 3 1 2 3 4 Njets [ p T > 100 GeV] 10−5 1.4 1.2 1 0.8 0.6 50 (c) 100 150 200 250 300 350 400 Subleading additional jet p T [GeV] (d) Figure 6.5: Results of αSISR tuning 2. Jet multiplicity distribution, Njets (with pjet T > 60 GeV), after tuning αSISR with the Njets > 3 bins (where jets predominantly originate from the parton shower) is used as input to Professor [47]. The unfolded CMS data are shown with total error bars. In each plot, the calculated distribution assuming the tuned αSISR is shown (solid line). The calculated distributions with the lower bound (dashed line) and the upper bound (dot-dashed line) of the tuned αSISR are also displayed. Beneath each plot is shown the ratio of theory predictions to data. The yellow bands indicate the total data uncertainty. 71 1/σvis dσvis /dNjets 1/σvis dσvis /dNjets b b 10−1 b CMS data 19.7fb−1 (8 TeV) arXiv:1510.03072 Data Uncertainty b b b b CMS data 19.7fb−1 (8 TeV) arXiv:1510.03072 Data Uncertainty b dummy b POWHEG+PYTHIA8 CUETP8M1 Theory/Data 1.4 1.2 1 0.8 0.6 2 3 4 b POWHEG+PYTHIA8 CUETP8M1 10−2 αISR S = 0.133 αISR S = 0.093 ISR αS = 0.142 10−3 Theory/Data 10−1 dummy 10−2 b b αISR S = 0.133 αISR S = 0.093 ISR αS = 0.142 b 1.4 1.2 1 0.8 0.6 5 6 Njets [ p T > 30 GeV] 0 1 2 b b b 10−1 CMS data 19.7fb−1 (8 TeV) arXiv:1510.03072 Data Uncertainty b POWHEG+PYTHIA8 CUETP8M1 αISR S = 0.133 αISR S = 0.093 αISR S = 0.142 b dummy 10−1 POWHEG+PYTHIA8 CUETP8M1 b 10−2 CMS data 19.7fb−1 (8 TeV) arXiv:1510.03072 Data Uncertainty αISR S = 0.133 αISR S = 0.093 αISR S = 0.142 b b 10−3 b b 10−3 1.4 1.2 1 0.8 0.6 0 1 10−4 b Theory/Data Theory/Data b dummy 10−2 4 5 Njets [ p T > 60 GeV] (b) 1/σvis dσvis /dp T [GeV−1 ] 1/σvis dσvis /dNjets (a) 3 1 2 3 4 Njets [ p T > 100 GeV] 10−5 1.4 1.2 1 0.8 0.6 50 (c) 100 150 200 250 300 350 400 Subleading additional jet p T [GeV] (d) Figure 6.6: Results of αSISR tuning 3. Jet multiplicity distribution, Njets (with pjet T > 30 GeV) and Subleading additional jet pT , after tuning αSISR with the Njets > 3 bins (where jets predominantly originate from the parton shower) are used as input to Professor [47]. The unfolded CMS data are shown with total error bars. In each plot, the calculated distribution assuming the tuned αSISR is shown (solid line). The calculated distributions with the lower bound (dashed line) and the upper bound (dot-dashed line) of the tuned αSISR are also displayed. Beneath each plot is shown the ratio of theory predictions to data. The yellow bands indicate the total data uncertainty. 72 6.5 The renormalisation scale µR From Sec. 6.4, upper and lower bound uncertainties for the optimal αSISR were found. It is also possible however, to vary αSISR by altering the renormalisation scale µR . Up to one loop correction, αS is given by αS ≈ 1 2 µ β(n) ln Λ2 R (6.2) QCD 33 − 2n (6.3) with β(n) = 12π with n being the number of active quark flavours and ΛQCD the QCD-scale. Using this relation between αS and µR , we can determine the scale variations corresponding to the uncertainties found for αSISR . This is displayed in Fig. 6.7. In Eq. 6.2, ΛQCD is first determined for αS set to the central value and µR set to 91.2 GeV (Z boson mass). Then the upper and lower bounds for αS are filled in and it is checked how much µR has to be varied for a fixed ΛQCD . The variations are displayed in the legend of Fig. 6.7 and correspond to 0.33 for the upper bound and 4.10 for the lower bound. αS(mZ) = 0.115 αS(mZ) = 0.096 αS(mZ) = 0.137 0.160 αS 0.140 0.120 0.100 0.080 101 0.33 · mZ mZ 102 µR (GeV) 4.10 · mZ 103 Figure 6.7: αS as a function of the renormalization scale, µR . The uncertainty on the tuned αS value (0.115) corresponds to variations of µR by factors of 0.33 for the upper bound and 4.10 for the lower bound. 73 6.6 Comparison between Professor Interpolation and MC Output 19.7 fb−1 (8 TeV) 1/σvis dσvis /dNjets CMS b b 10−1 b Data αISR S = 0.115 (Interpolation) αISR S = 0.115 (MC) b Theory/Data 10−2 b b 1.4 1.2 1 0.8 0.6 2 3 4 5 6 Njets [ p T > 30 GeV] Figure 6.8: Comparison between the Professor interpolation and the MC output for tuning result 1 with αSISR = 0.115 for the Njets (pT > 30 GeV) distribution. The interpolated results from Professor (red) nicely overlap with the MC data (blue) generated for the same value of αSISR . The yellow bands denote the data uncertainties. The Professor tuning result 1 is given by αSISR = 0.115 and is used as input to a POWHEG+PYTHIA8 MC run. The Professor interpolation and MC output are compared for the Njets (pT > 30 GeV) observable as shown in Fig. 6.8. It is to be seen that both results are nicely in agreement with each other, which provides confidence that Professor indeed produces correctly interpolated observable distributions. 74 6.7 Validation with other Top Quark Distributions The studied tunes are validated with the TOP-15-011 distributions (with Rivet routine CMS 2015 I1370682) [60]. Concretely, it is checked using tune results 1, 2 and 3 (see Fig. 6.3) what the MC output with respect to data is for the observables ptT , ∆φtt̄ , ptTt̄ and mtt̄ in the single lepton and dilepton channel. In the single lepton and dilepton channel, we can see from Figs. 6.9 (a), (b), (c) and (d) that there is a good agreement between the different αSISR values. However, as expected, the ptT remains unchanged and is still not well described by the MC6 . Figs 6.10 (a) and (b) show how for the ptTt̄ and mtt̄ distributions higher values of αSISR are slightly preferred over tune 1. From Figs. 6.10 (c) and (d), the variations in αSISR in the ptTt̄ and mtt̄ distributions do not seem to have a significant effect on the MC output. 6 The incorrect description of top pT is known to be caused by missing higher order effects. 75 tt̄ → bblνjj | 19.7 fb−1 (8 TeV) 0.007 Data = 0.115 (CMS Tune 1) αISR S = 0.104 (CMS Tune 2) αISR S = 0.113 (CMS Tune 3) αISR S = 0.121 (ATLAS Tune) αISR S = 0.127 (CUETP8M1) αISR S = 0.137 (CTEQ5M) αISR S b b 0.006 0.005 b 0.004 b 0.003 tt̄ → bblνjj | 19.7 fb−1 (8 TeV) CMS (1/σ ) dσ/d∆φ (1/σ ) dσ/dpT [GeV−1 c] CMS Data = 0.115 (CMS Tune 1) αISR S = 0.104 (CMS Tune 2) αISR S = 0.113 (CMS Tune 3) αISR S = 0.121 (ATLAS Tune) αISR S = 0.127 (CUETP8M1) αISR S = 0.137 (CTEQ5M) αISR S b 1 b b b b 0.002 10−1 0.001 b b b b Theory/Data Theory/Data 0 1.4 b 1.2 1 0.8 0.6 0 100 200 300 1.4 1.2 1 0.8 0.6 400 500 pTt [GeV/c] 0 0.5 1 1.5 (a) Data = 0.115 (CMS Tune 1) αISR S = 0.104 (CMS Tune 2) αISR S = 0.113 (CMS Tune 3) αISR S = 0.121 (ATLAS Tune) αISR S = 0.127 (CUETP8M1) αISR S = 0.137 (CTEQ5M) αISR S b b 0.006 0.005 b 0.004 0.003 b 0.002 b 1 Data = 0.115 (CMS Tune 1) αISR S = 0.104 (CMS Tune 2) αISR S = 0.113 (CMS Tune 3) αISR S = 0.121 (ATLAS Tune) αISR S = 0.127 (CUETP8M1) αISR S = 0.137 (CTEQ5M) αISR S b b b 10−1 0.001 b b b 0 1.4 Theory/Data Theory/Data 3 ∆φtt̄ [rad] tt̄ → bblνlν | 19.7 fb−1 (8 TeV) CMS (1/σ ) dσ/d∆φ (1/σ ) dσ/dpT [GeV−1 c] 0.007 2.5 (b) tt̄ → bblνlν | 19.7 fb−1 (8 TeV) CMS 2 1.2 1 0.8 0.6 0 50 100 150 200 250 300 350 400 pTt [GeV/c] (c) 1.4 1.2 1 0.8 0.6 0 0.5 1 1.5 2 2.5 3 ∆φtt̄ [rad] (d) Figure 6.9: Validation of the TOP-15-011 distributions for the variables ptT and ∆φtt̄ in the single lepton ((a) and (b)) and the dilepton ((c) and (d)) channels for the four different tune results. The ATLAS ATTBAR-POWHEG tune and the PYTHIA8 (CTEQ5M) and PYTHIA6 (CUETP8M1) defaults are also displayed. 76 tt̄ → bblνjj | 19.7 fb−1 (8 TeV) (1/σ ) dσ/dpTtt̄ [GeV−1 c] 0.016 Data = 0.115 (CMS Tune 1) αISR S = 0.104 (CMS Tune 2) αISR S = 0.113 (CMS Tune 3) αISR S = 0.121 (ATLAS Tune) αISR S = 0.127 (CUETP8M1) αISR S = 0.137 (CTEQ5M) αISR S b 0.014 b 0.012 b 0.01 0.008 b 0.006 tt̄ → bblνjj | 19.7 fb−1 (8 TeV) CMS (1/σ ) dσ/dmtt̄ [GeV−1 c2 ] CMS 0.004 Data = 0.115 (CMS Tune 1) αISR S = 0.104 (CMS Tune 2) αISR S = 0.113 (CMS Tune 3) αISR S = 0.121 (ATLAS Tune) αISR S = 0.127 (CUETP8M1) αISR S = 0.137 (CTEQ5M) αISR S b 10−2 b b b 10−3 b b 10−4 b b 0.002 10−5 b b Theory/Data Theory/Data 0 1.4 1.2 1 0.8 0.6 0 50 100 150 200 b 1.4 1.2 1 0.8 0.6 250 300 pTtt̄ [GeV/c] 400 600 800 1000 (a) b b 0.014 0.012 0.01 0.008 b Data = 0.115 (CMS Tune 1) αISR S = 0.104 (CMS Tune 2) αISR S = 0.113 (CMS Tune 3) αISR S = 0.121 (ATLAS Tune) αISR S = 0.127 (CUETP8M1) αISR S = 0.137 (CTEQ5M) αISR S 0.006 tt̄ → bblνlν | 19.7 fb−1 (8 TeV) CMS (1/σ ) dσ/dmtt̄ [GeV−1 c2 ] (1/σ ) dσ/dpTtt̄ [GeV−1 c] 0.016 1400 1600 mtt̄ [GeV/c2 ] (b) tt̄ → bblνlν | 19.7 fb−1 (8 TeV) CMS 1200 b 10−2 b b b 10−3 b 10−4 Data = 0.115 (CMS Tune 1) αISR S = 0.104 (CMS Tune 2) αISR S ISR αS = 0.113 (CMS Tune 3) = 0.121 (ATLAS Tune) αISR S = 0.127 (CUETP8M1) αISR S = 0.137 (CTEQ5M) αISR S b 0.004 10−5 b 0 1.4 b b Theory/Data Theory/Data 0.002 1.2 1 0.8 0.6 0 50 100 150 200 250 300 pTtt̄ [GeV/c] (c) 1.4 1.2 1 0.8 0.6 400 600 800 1000 1200 1400 1600 mtt̄ [GeV/c2 ] (d) Figure 6.10: Validation of the TOP-15-011 distributions for the variables ptTt̄ and mtt̄ in the single lepton ((a) and (b)) and the dilepton ((c) and (d)) channels for the four different tune results. The ATLAS ATTBAR-POWHEG tune and the PYTHIA8 (CTEQ5M) and PYTHIA6 (CUETP8M1) defaults are also displayed. 77 6.8 Summary and Discussion In this study, the PYTHIA8 parameter αSISR is tuned using Njets -distributions. The optimal value is chosen to be αSISR = 0.115+0.021 −0.019 as its correspondence with data is the best as can be seen from Figs. 6.4 (a), (b), (c) and (d) and it has the lowest uncertainties. Furthermore, within uncertainties it is in agreement with the ATLAS ATTBARPOWHEG tune (Fig. 6.3 and ref. [59]). The uncertainties seem to correspond to a factor 0.33 and 4.10 for the respective upper and lower bounds of the renormalisation scale µR . The different tunes give αSISR of the order 0.10-0.11, however when applying these to TOP-15-011 distributions (Figs 6.9 and 6.10), the value of αSISR = 0.1273 seems to give a slightly better description. For further tests, the tt̄ samples for ICHEP 2016 have been submitted with αSISR = 0.1273, with the same value as the PYTHIA6 default (CTEQ5) used in Run I CMS simulations. CUETP8M1 predictions using underlying event Drell-Yan pT and rapidity y, are tested with the tuned αSISR values. No significant changes have been observed [61]. The main differences have been observed in underlying event observables up to ∼10-15% [61]. This shows that it is desirable to have a new underlying event tune that has a lower αSISR value. Tuning αSISR in POWHEG+PYTHIA8 is the most straightforward approach to fix the discrepancies seen in the Njets distributions (e.g. Fig. 6.1). This process however may not be as simple in MADGRAPH5 aMC@NLO+PYTHIA8 since for this case the αSISR must be modified in the matrix element as well. The tune found makes the agreement of 13 TeV data with POWHEG+PYTHIA8 as good as 8 TeV with MADGRAPH5+PYTHIA6, i.e. a very good description of all distributions (in the considered low pT and mass ranges) except for ptT , which is due to missing higher order effects. Note also that parton shower αSISR tuning should depend on the QCD scale choice and on the hdamp parameter in POWHEG (which is taken to be hdamp = mt ). In a finer tuning attempt hdamp and αSISR should be tuned simultaneously or consecutively. For instance, one can first tune αSISR as has been done in this study and then tune the hdamp variable using the first additional jet pT distribution as input. To conclude with, the description of Njets is fixed using a lower αSISR value than the PYTHIA8 default of 0.1365. This change does not cause significant alterations and therefore with the lower αSISR value all top quark distributions except ptT are described well by POWHEG+PYTHIA8. These results are also made public as additional material to [58]. 78 Hoofdstuk 7 Nederlandstalige Samenvatting 7.1 Introductie Het Standaardmodel (SM) van de elementaire deeltjesfysica is een zeer succesvolle theorie die in staat is om een erg rijk gamma aan experimenten te beschrijven. Het beschrijft de drie fundamentele krachten van het elektromagnetisme, de zwakke kernkracht en de sterke kernkracht, zoals beschreven in Sectie 1.2. De vierde fundamentele kracht, nl. de zwaartekracht, past niet binnen dit framework, vermits het SM niet compatibel is met de algemene relativiteitstheorie. Daarnaast levert het een classificatie van alle fundamentele deeltjes die tot op heden gekend en waargenomen zijn. Deze worden beschreven in Sectie 1.1. Het waarnemen van deze fundamentele deeltjes gebeurt d.m.v. een groot aantal experimenten verspreid over de ganse wereld. In deze studie wordt toegespitst op het CMS (Compact Muon Solenoid ) experiment (hoofdstuk 3), uitgevoerd in CERN in Genève, Zwitserland. CMS is een detector die geı̈nstalleerd is op de LHC (Large Hadron Collider ), waarmee protonen versneld worden tot enorme snelheden en botsen op welbepaalde interactiepunten aan hoge massacentrumenergieën. CMS is cilindrisch van vorm en bevindt zich rond een dergelijk interactiepunt waar het tracht de vervalproducten van de botsingen te detecteren. Hoewel het SM erg succesvol is, bestaan er toch vele tekortkomingen en fenomenen die niet verklaard kunnen worden zonder een uitbreiding van de theorie. Enkele van deze tekortkomingen worden gegeven in Sectie 1.4. Een erg populaire theorie die het SM uitbreidt is Supersymmetrie (SUSY), zoals uiteengezet in hoofdstuk 2. SUSY postuleert het bestaan van een superpartner voor alle reeds gekende fundamentele deeltjes. 79 Hier wordt het Minimale Supersymmetrische Model (MSSM) aangenomen. Dit betreft een minimale uitbreiding van het SM en veronderstelt het behoud van R-pariteit. Deze symmetrie zorgt ervoor dat supersymmetrische deeltjes steeds in paren geproduceerd worden en dat er een lichtst supersymmetrisch deeltje (LSP) bestaat, het neutralino χ01 , dat bovendien stabiel is. Bovenstaande inzichten worden in Sectie 2.3 in meer detail uiteengezet. Een SUSY-onderzoek dat in deze thesis gedaan wordt, staat hieronder in Sectie 7.2 beschreven. Naast de uitbreiding van het SM d.m.v. SUSY, is het in deze thesis ook belangrijk om stil te staan bij Monte Caro (MC) generatoren. Deze zijn onmisbaar bij hoge energie fysica experimenten. MC generatoren bestaan uit een set van modellen die instaan voor het simuleren van evenementen zoals deze die optreden bij botsingen tussen hadronen (of hier, botsingen tussen protonen). Wij spitsen ons toe op twee aparte types van MC generatoren. Ter illustratie bekijken wij POWHEG en PYTHIA. POWHEG is een Matrix Element (ME) MC generator. Dit betekent dat deze instaat voor het genereren van het initiële hard scatter proces waarbij een hoge momentumoverdracht plaatsvindt. Dit is m.a.w. de interactie tussen twee partonen uit elk van beide protonen. Powheg berekent dit proces tot op Next-to-Leading-Order (NLO), waarbij het dus een harde partonenemissie in rekening neemt, hetgeen weergegeven wordt in het Feynman diagram in Fig. 4.2. PYTHIA is een MC generator die hoge energie fysica evenementen genereert. Dit zijn sets van deeltjes die geproduceerd werden uit de interactie tussen twee inkomende deeltjes. Het bevat theoretische modellen die de evolutie van een proces met hoge momentumoverdracht simuleren tot een complexe multi-hadronische finale toestand. Deze evolutie wordt Parton Shower (PS) genoemd. Een parton verliest immers steeds momentum tijdens diens propagatie doorheen de ruimte (d.m.v. de interacties met quarks en gluonen). Eens een bepaalde drempelwaarde overschreden wordt, treedt hadronisatie op waarbij kleurloze finale toestanden gevormd worden. Aangezien MC generatoren tal van benaderingen gebruiken, worden parameters scherpgesteld om er voor te zorgen dat de predicties overeenkomen met wat gemeten wordt in data. Dit proces heet tuning en wordt gedaan met specifieke programma’s. Hiertoe kan Professor gebruikt worden, een tuning code die in meer detail beschreven staat in Appendix A. Een concrete toepassing van het scherpstellen van een PYTHIA8 parameter, nl. αSISR , wordt uiteengezet in Sectie 7.3, verder in deze tekst. 80 7.2 Deel I: Zoeken naar scalaire top quark partners Deel I van deze studie is een deelonderzoek in een SUSY-analyse m.b.t. het √zoeken naar de scalaire superpartner van de top quark bij een massacentrumenergie s = 13 TeV voor proton-proton botsingen. Het gepostuleerde proces wordt gegeven in Figuur 5.1, waarbij specifiek wordt gezocht naar het kanaal waarin beide W bosonen leptonisch vervallen. Het is de bedoeling bij dergelijke studies dat een signaalregio geconstrueerd wordt. Dit is een bepaald stuk van de faseruimte waar het signaal de achtergrondprocessen gegeven op blz. 56 domineert. De discriminerende parameter die hiervoor gebruikt wordt is MT 2 (ll) (gedefinieerd door Vgl. 5.1). Het is uiteraard belangrijk dat in de signaalregio de achtergronden, die berekend zijn door Monte Carlo generatoren, correct opgenomen zijn. Zoals aangegeven in Tabel 5.1 en Figuur 5.4, wordt opgemerkt dat de tt̄ + jets achtergrond toch voor hoge waarden van MT 2 (ll) (> 140 GeV/c2 ) aanwezig is. Voor een dergelijk SM proces is dit onmogelijk. Een verklaring en oplossing voor het optreden van tt̄ + jets evenementen in dit gebied wordt vervolgens uit de doeken gedaan. In de regio MT 2 (ll) > 140 GeV/c2 worden de aanwezige tt̄ + jets evenementen specifiek uitgezocht. De gereconstrueerde kinematische informatie (RECO) wordt gelinkt aan de gegenereerde kinematische informatie (GEN) om te achterhalen wat de moederdeeltjes zijn van de geselecteerde leptonen. Een voorbeeld hiervan wordt gegeven in Sectie 5.4. Van de 37 gevonden evenementen blijken er 29 afkomstig van b quark verval en niet van leptonisch W boson verval. Dit betekent dat een deel van de tt̄ + jets evenementen verkeerdelijk geselecteerd worden in de analyse. Voor dit probleem worden twee oplossingen vooropgesteld. Ten eerste wordt gekeken of een RelIso04 < 0.12 cut het probleem zou oplossen. De motivatie voor deze keuze is om de grootte van de fictieve kegel getekend rondom het geselecteerde lepton te vergroten om zo meer jet activiteit op te vangen en zo leptonen van b quark verval (dicht bij jet activiteit) te verwijderen. Dit kan eenvoudig begrepen worden uit Vgl. 5.3 en Figuur 5.5. Het effect hiervan was dat deze methode wel degelijk ∼80% van de verkeerdelijk geselecteerde tt̄ + jets evenementen reduceert, maar dit echter ook inefficiënt is voor het signaal. De methode bestaat er in om een MultiIsoV2 cut toe te passen, zoals gedefinieerd in Vgl. 5.4. Deze heeft het gewenste effect om wederom ∼80% van de verkeerdelijk geselecteerde leptonen uit de hoge MT 2 (ll) regio te verwijderen. Dit is 81 weergegeven in de Figuren 5.6b en 5.7. Daarnaast is het toepassen van deze cut zeer signaalefficiënt, vermits deze slechts een verlies van 4% oplevert. 7.3 Deel II: Scherpstellen van αISR in POWHEG+ S PYTHIA8 voor het modelleren van top quark paren Deel II van deze studie situeert zich rond het zoeken naar oplossingen voor een gekend probleem bij het genereren van tt̄ evenementen m.b.v. de MC generatoren POWHEG+ PYTHIA8. In Fig. 6.1 wordt aangetoond hoe het resultaat van de simulatie in functie van jet-multipliciteit Njets duidelijk afwijkt van wat gemeten wordt in data. Een oplossing wordt hiertoe voorgesteld waarbij een specifieke parameter in PYTHIA8, nl. SpaceShower: alphaSvalue, wordt scherpgesteld. Deze wordt in het vervolg van de tekst aangeduid met αSISR . Het scherpstellen gebeurt d.m.v. Professor, een programma daartoe speciaal ontworpen. Het scherpstellen wordt in het vervolg tuning genoemd. Er worden 13 MC runs gegenereerd voor verschillende waarden van αSISR zoals gegeven in Tabel 6.1. Voor deze runs worden drie tunings uitgevoerd naar een bepaalde observabele (of observabelen) zoals aangeduid in Tabel 6.2. Dit gebeurt aan de hand van een χ2 methode die in meer detail in Appendix A wordt toegelicht. Elk tuning-resultaat staat samengevat in Tabel 6.3 met bijhorende onzekerheden. Het effect van elke tuning poging op de verschillende observabelen wordt weergegeven in Figuren 6.4, 6.5 en 6.6. Zij worden bovendien vergeleken met de ATLAS ATTBARPOWHEG tune [59], de PYTHIA8 (CUETP8M1) en PYTHIA6 (CTEQ5M) standaarden in Fig 6.3. Hieruit wordt besloten dat tune 1 de meest optimale is vermits deze met het ATLAS-resultaat en de PYTHIA6 en PYTHIA8 standaarden in overeenkomst is, de kleinste onzekerheden heeft en de distributies in Fig. 6.4 accuraat beschrijft. Voor de optimale tune 1 wordt voor diens onzekerheden nagegaan met welke onzekerheden in de renormalisatieschaal deze corresponderen. Dit wordt grafisch weergegeven in Fig. 6.7. De onzekerheden corresponderen met een variatie van de renormalisatieparameter volgens de factoren 0.33 en 4.10. Toepassing van de gevonden tunes op TOP-15-011 distributies [60] is weergegeven in Figuren 6.9 en 6.10. Hier worden de MC runs voor de verschillende tunes in functie 82 van de observabelen ptT , ∆φtt̄ , ptTt̄ en mtt̄ uitgezet. Ten eerste valt hierbij op dat ptT uitermate slecht beschreven wordt. Dit is te wijten aan hogere orde effecten die niet in rekening gebracht zijn en niet opgelost kunnen worden door tuning. Ten tweede valt op dat de MC uitkomsten voor alle tunes voor de overige distributies erg gelijkaardig zijn met elkaar. Op het zicht leren we hier dat de beste overeenkomsten toch bij de hogere waardan van αSISR te vinden zijn. Daarom werden de tt̄ samples voor ICHEP 2016 aangevraagd met de waarde van αSISR = 0.1273, hetgeen dezelfde waarde is van de PYTHIA6 standaard en dus iets hoger ligt dan de eigen gevonden tunes van orde 0.10-0.11. Voor een toekomstige poging tot tuning, kan men proberen om αSISR gelijktijdig of opeenvolgend te tunen met de hdamp parameter in POWHEG daar dit gunstige resultaten levert volgens een studie gedaan door de ATLAS collaboratie [59]. Men kan concluderen dat in deze studie de beschrijving van de Njets observable hersteld is door het gebruiken van een lagere αSISR waarde dan de standaardwaarde van 0.1365 voor PYTHIA8. Deze aanpassing veroorzaakt geen significante veranderingen voor de overige top quark distributies (ptT , ∆φtt̄ , ptTt̄ en mtt̄ ) daar deze nog steeds accuraat beschreven worden door POWHEG+PYTHIA8 behalve ptT . Deze resultaten zijn ook publiek gemaakt als extra materiaal voor [58]. 83 Appendix A Professor: A tuning tool for Monte Carlo event generators Professor (PROcedure For EStimating Systematic errORs) is a parametrisation-based tuning tool. This means that the approach taken is to parametrise the generator behaviour by fitting a polynomial to the generator response (here referred to as MCb ) for each observable bin to changes in the P -element parameter vector p = (p1 , ..., pP ). P denotes the amount of parameters that can be changed in the event-generator. One uses a polynomial of second order to do the bin parametrisation. Say that p 0 is a randomly chosen point in parameter space (in between the bounds of a hypercube determined by the user), then the polynomial has the following form: (b) M Cb (p) ≈ α0 + P X i (b) βi p0i + P X (b) γij p0i p0j (A.1) i≤j with p 0 = p - p 0 . The left side of the above equations denotes the output from the event-generator at parameter point p, whereas the right side contains the unknown coefficients [α, β, γ]. The amount of coefficients to be determined depends on the order of the polynomial (here: 2) and the amount of parameters P to be tuned, also referred to as the dimension of the parameter space. For a 2nd order fit, it can be readily computed from Eq. A.1: (P ) N2 = 1 + P + P (P + 1)/2 (A.2) Once we have a set of functions MCb , we have a very fast way of predicting the gen84 erator behaviour at a certain point in parameter space p, as it is not necessary to run the generator again. This allows one to construct a goodness of fit function which compares the data to the Monte Carlo, and thus to come up with an optimal point p best of the tuned parameters, even though p best was not specifically used in the computation. A.1 Determining the response function MCb(p) Given sets of sampled points {p} and generator values {M Cb (p)}, eq. A.1 can be transformed into a matrix equation, which is acquired for every bin and for every observable: e (b) v(b) = Pc (A.3) e contains the variations of the parameter points (p - p 0 ) for different orders. We get P in matrix notation, if we presume 2 parameters such that p = (px , py ) and N evaluated parameter points for a certain bin b, that: α0 βx 1 x1 y1 x21 x1 y1 y12 v1 v 2 1 x2 y 2 x2 x2 y 2 y 2 βy 2 2 (A.4) · .. = .. . γxx . γxy 2 1 xN yN x2N xN yN yN vN {z } | {z } | γyy e v (values) | {z } P(sampled parameter sets) c (coefficients) with the values vi being the generator computed value. This can be understood if we simply look at the equation we get for the first row. With x denoting the first parameter and y denoting the second parameter, we get: v1 = α0 + x1 βx + y1 βy + x21 γxx + x1 y1 γxy + y12 γyy (A.5) with x1 = p0x = px −px0 and y1 = p0y = py −py0 both evaluated at point n◦ 1 in parameter space. This is exactly eq. A.1. e known, we still need to compute c(b) . This can be done With the matrices v(b) and P e by a so-called pseudo-inversion Ie of P: 85 e (b) e P]v c(b) = I[ (A.6) Pseudo-inversion is comparable to regular (n×n)-matrix inversion, but here for (m×n)matrices. A.2 Goodness of Fit (GoF) In the previous section, the functions MCb (p) were determined, giving us the generator response for every point in parameter space for every observable bin. What remains now, is to compute a goodness of fit function that is minimised for a certain parameter point p best , which can be found from extrapolating the GoF function to a minimum. It is possible to add per-observable weights wO for each observable O. We choose a χ2 -function defined according to: 2 χ (p) = X O X (MCb (p) − Rb )2 wO ∆2b b ∈ O (A.7) where Rb is the data value for bin b, and the error ∆b is calculated from the sum in quadrature of the MC-error and the error in the data. It should be noted that determination of the weights is a subjective process and should be chosen according to criteria set by the user. A.3 Error Estimation: Construction of Eigentunes Once an optimal set of parameters p best is found, it remains to find the correct errors on the acquired tunes. The upper and lower bounds we refer to as the eigentunes. For simplicity we now consider a two dimensional parameter space p = (px ,py ), with optimal tune values (p∗x ,p∗y ). The eigentunes can be constructed by considering a Taylor’s expansion around the χ2 (p). 2 2 χ (px , py ) ≈ χ (p∗x , p∗y ) 2 2 2 2 1 1 ∗ 2∂ χ ∗ 2∂ χ + (px − px ) + (py − py ) 2 ∂p2x (p∗x ,p∗y ) 2 ∂p2y (p∗ ,p∗ ) x y 2 2 ∂ χ + (px − p∗x )(py − p∗y ) ∂px ∂py (p∗x ,p∗y ) 86 (A.8) If we compare this formula to the general equation of a rotated ellipse centred about (a,b) (x − a)2 (y − b)2 xy + + 2, (A.9) A2 B2 C with A and B the lengths of the axes and B14 − 4 A12 C12 < 01 , we learn that for a specific value of ∆χ2 = χ2 (px , py ) - χ2 (p∗x , p∗y ) an ellipse in parameter space can be drawn. The standard Professor prescription is to choose ∆χ2 = χ2 (p∗x , p∗y ). This means the minimum of the χ2 is shifted up by the value of χ2 (p∗x , p∗y ). The outer edges of the ellipse are then taken to be the errors on the tunes as depicted in Fig. A.1 1= Figure A.1: Ellipse denoting equal values of the χ2 in parameter space. 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